THE  LIBRARY 

OF 

THE  UNIVERSITY 
OF  CALIFORNIA 

LOS  ANGELES 


u 
NOV   7       1957 


INTERNATIONAL  CHEMICAL  SERIES 
H.  P.  TALBOT,  PH.D.,  Sc.D.,  CONSULTING  EDITOR 


FLUIDITY  AND  PLASTICITY 


PUBLISHERS     OF     BOOKS     FOIO 

Electrical  \Xbrld  v  Engineering  News -Record 
Power  v  Engineering  and  Mining  Journal-Press 
Chemical  and  Metallurgical  Engineering 
Electric  Railway  Journal  v  Coal  Age 
American  Machinist  v  Ingenieria  International 
Electrical  Merchandising  v  BusTransportation 
Journal  of  Electricity  and  Western  Industry 
Industrial  Engineer 


(Frontispiece). 


FLUIDITY  AND  PLASTICITY 


BY 

EUGENE  C.  BINGHAM,  PH.D. 

PROFESSOR   OF   CHEMISTRY  AT  LAFAYETTE   COLLEGE,    EASTO-V,    PENNSYLVANIA 


FIRST  EDITION 
SECOND  IMPRESSION 


McGRAW-HILL  BOOK  COMPANY,  INC. 
NEW  YORK:  370  SEVENTH  AVENUE 

LONDON:  6  &  8  BOUVERIE  ST.,  E.  C.  4 

1922 


COPYRIGHT,  1922,  BY  THE 
MCGRAW-HILL  BOOK  COMPANY,  INC. 


PRINTED   IN   THE    UNITED   STATES   OF   AMERICA 


All  translation  rights  reserved 


Physics 
Library 

qc 
ni 


To  my  sister 


919548 


PREFACE 

Our  knowledge  of  the  flow  of  electrical  energy  long  ago  de- 
veloped into  the  science  of  Electricity  but  our  knowledge  of  the 
flow  of  mailer  has  even  yet  not  developed  into  a  coordinate 
science.  In  this  respect  the  outcome  of  the  labors  of  the  hydro- 
dynamicians  has  been  disappointing.  The  names  of  Newton, 
Navier,  Poisson,  Graham,  Maxwell,  Stokes  and  Helmholtz  with 
a  thousand  others  testify  that  this  field  has  been  well  and  com- 
petently tilled.  Even  from  the  first  the  flow  of  liquids  has  been 
a  subject  of  practical  importance,  yet  the  subject  of  Hydraulics 
has  never  become  more  than  an  empirical  subject  of  interest 
merely  to  the  engineer. 

Unfortunately  the  theory  is  complicated  in  that  the  flow  of 
matter  may  be  hydraulic  (turbulent),  viscous  (linear),  or  plastic, 
dependent  upon  the  conditions.  It  was  in  1842  that  viscous 
flow  was  first  differentiated  from  hydraulic  flow,  and  only  now 
are  we  coming  to  realize  the  important  distinction  between  vis- 
cous and  plastic  deformation.  Considering  the  confusion  which 
has  existed  in  regard  to  the  character  of  flow,  it  is  not  surprising 
that  there  has  been  uncertainty  in  regard  to  precise  methods  of 
measurement  and  that  exact  methods  have  been  discovered,  only 
to  be  forgotten,  and  rediscovered  independently  later.  As  a 
result,  the  amount  of  really  trustworthy  data  in  the  literature 
on  the  flow  of  matter  under  reproducible  conditions  is  limited, 
often  to  an  embarrassing  extent. 

If  we  are  to  have  a  theory  of  flow  in  general,  we  must  consider 
matter  in  its  three  states.  No  such  general  theory  has  appeared, 
although  one  is  manifestly  needed  to  give  the  breath  of  life  to 
the  dead  facts  about  flow.  The  author  offers  the  theory  given 
in  the  following  pages  with  the  utmost  trepidation.  Although 
he  has  given  several  years  to  the  pleasant  task  of  supporting  its 
most  important  conclusions,  a  lifetime  would  be  far  too  short  to 
complete  the  work  unaided.  The  author  makes  no  apology  for 
any  lack  of  finality.  Parts  of  the  theory  which  have  already 


x  PREFACE 

found  their  way  into  print  have  awakened  a  vigorous  discussion 
which  is  still  in  progress.  This  is  well,  for  our  science  thrives  on 
criticism  and  through  the  collaboration  of  many  minds  the  final 
theory  of  flow  will  be  evolved. 

Without  going  considerably  beyond  the  limits  which  we  have 
placed  upon  ourselves,  it  is  impossible  to  refer  even  briefly  to  all 
of  the  important  papers  on  the  subject.  References  given  in 
the  order  that  they  come  up  in  the  discussion  are  not  the  best 
suited  for  later  reference.  The  novel  plan  has  been  tried  of 
placing  nearly  all  of  our  references  in  a  separate  appendix  which 
is  also  an  author  index  and  is,  therefore,  arranged  alphabetically 
under  the  authors'  names.  In  the  text  the  name  of  the  author 
and  the  year  of  publication  of  the  monograph  is  usually  sufficient 
for  our  purpose,  but  sometimes  the  page  is  also  added.  The  titles 
of  the  monographs  are  usually  given  in  the  hope  that  this  bibli- 
ography may  be  of  considerable  service  to  investigators  who  are 
looking  up  a  particular  line  of  work  connected  with  this  general 
subject. 

It  is  a  pleasure  to  thank  Dr.  R.  E.  Wilson  of  the  Massachusetts 
Institute  of  Technology  and  Dr.  Hamilton  Bradshaw  of  the  E.  I. 
bu  Pont  de  Nemours  &  Company  for  reading  over  the  manu- 
script and  Dr.  James  Kendall  for  examining  the  proof.  Profes- 
sor Brander  Matthews  of  Columbia  University,  Professor  James 
Tupper  and  Professor  James  Hopkins  of  Lafayette  College  have 
assisted  in  important  details.  The  author  gladly  acknowledges 
the  valuable  assistance  of  his  colleagues  and  co-workers,  Dr. 
George  F.  White,  Dr.  J.  Peachy  Harrison,  Dr.  Henry  S.  Van 
Klooster,  Mr.  Walter  G.  Kleinspehn,  Mr.  Henry  Green,  Mr. 
William  L.  Hyden,  Mr.  Landon  A.  Sarver,  Mr.  Delbert  F.  Brown, 
Mr.  Wilfred  F.  Temple,  Mr.  Herbert  D.  Bruce,  and  others. 

The  author  is  especially  indebted  to  the  University  of  Rich- 
mond for  the  leisure  which  made  possible  a  considerable  portion 
of  this  work. 

EUGENE  C.  BINGHAM. 

EASTON,  PA. 

Feb.  11,  1922. 


CONTENTS 


PAGE 
PREFACE vii 

Part  I.  Viscometry 

,  CHAPTER 

I.  PRELIMINARY.     METHODS  OP  MEASUREMENT 1 

II.  THE  LAW  OP  POISEUILLE 8 

III.  THE  AMPLIFICATION  OP  THE  LAW  OF  POISEUILLE 17 

IV.  Is  THE  VISCOSITY  A  DEFINITE  PHYSICAL  QUANTITY? 58 

V.  THE  VISCOMETER 62 

Part  II.  Fluidity  and  Plasticity  and  Other  Physical  and  Chemical  Properties 

I.  VISCOSITY  and  FLUIDITY 81 

II.  FLUIDITY  AND  THE  CHEMICAL  COMPOSITION  AND  CONSTITUTION 

OF  PURE  LIQUIDS 106 

III.  FLUIDITY    AND    TEMPERATURE,    VOLUME,    PRESSURE.     COLLI- 

SIONAL    AND  DlFFUSIONAti  VISCOSITY 127 

IV.  FLUIDITY  AND  VAPOR  PRESSURE 155 

V.  THE  FLUIDITY  OF  SOLUTIONS 160 

VI.  FLUIDITY  AND  DIFFUSION 188 

VII.  COLLOIDAL  SOLUTIONS 198 

VIII.  THE  PLASTICITY  OF  SOLIDS 215 

IX.  THE  VISCOSITY  OF  GASES 241 

X.  SUPERFICIAL  FLUIDITY 254 

XI.  LUBRICATION 261 

XII.  FURTHER  APPLICATIONS  OP  THE  VISCOMETRIC  METHOD 279 

APPENDIX  A.  PRACTICAL  VISCOMETRY 296 

APPENDIX  B.  PRACTICAL  PLASTOMETRY 320 

APPENDIX  C.  TECHNICAL  VISCOMETERS 324 

APPENDIX  D.  MEASUREMENTS  OF  POISEUILLE 331 

VISCOSITIES  AND  FLUIDITIES  OF  WATER  FLUIDITIES  OF  ETHYL  ALCO- 
HOL AND  SUCROSE  SOLUTIONS 341 

RECIPROCALS >342 

FOUR-PLACE  LOGARITHMS 345 

BIBLIOGRAPHY  AND  AUTHOR  INDEX 347 

SUBJECT  INDEX.  .   431 


FLUIDITY  AND  PLASTICITY 

PART  I 
VISCOMETRY 

CHAPTER  I 
PRELIMINARY.     METHODS  OF  MEASUREMENT 

Introductory. — What  one  may  be  pleased  to  call  "dominant 
ideas"  have  so  stimulated  the  work  on  viscosity,  that  it  would 
be  entirely  possible  to  treat  the  subject  of  viscosity  by  consider- 
ing in  turn  these  dominant  ideas. 

Practically  no  measurements  from  which  viscosities  may  be 
calculated  were  made  prior  to  1842,  yet  very  important  work 
was  being  done  in  Hydrodynamics,  and  the  fundamental  laws 
of  motion  were  established  during  this  preliminary  period.  To 
this  group  of  investigations  belong  the  classical  researches  of 
Bernouilli  (1726),  Euler  (1756),  Prony  (1804),  Navier  (1823), 
and  Poisson  (1831).  In  the  development  of  Hydrodynamics 
much  experimental  work  was  done  upon  the  flow  of  water  in 
pipes  of  large  bore  by  Couplet  (1732),  Bossut  (1775),  Dubuat 
(1786),  Gerstner  (1800),  Girard  (1813),  Darcy  (1858),  but  this 
work  could  not  lead  to  the  elucidation  of  the  theory  of  viscosity 
as  we  shall  see.  Important  work  belonging  to  this  preliminary 
period  was  also  done  by  Mariotte  (1700),  Galileo  (1817),  S'Grave- 
sande  (1719),  Newton  (1729),  D'Alembert  (1770),  Boscovich 
(1785),  Coulomb  (1801),  Eytelwein,  (1814). 

It  is  to  Poiseuille  (1842)  that  we  owe  our  knowledge  of  the 
simple  nature  of  flow  in  capillary  spaces,  which  is  in  contrast 
with  the  complex  condition  of  flow  in  wide  tubes,  heretofore 
used.  He  wished  to  understand  the  nature  of  the  flow  of  the 
blood  in  the  capillaries,  being  interested  in  internal  friction  from 
the  physiological  point  of  view.  He  made  a  great  many  meas- 

1 


2  FLUIDITY  AND  PLASTICITY 

urements  of  the  rates  of  flow  of  liquids  through  capillary  tubes, 
which  are  still  perhaps  unsurpassed.  They  lead  directly  to  the 
laws  of  viscous  resistance  and  they  will  be  described  in  detail 
in  a  later  chapter.  The  theoretical  basis  for  these  laws  and  a 
definition  of  viscosity  were  supplied  by  the  labors  of  Hagen 
(1854),  G.  Wiedemann  (1856),  Hagenbach  (1860),  Helmholtz 
(I860),  Maxwell  (1860).  Since  the  velocity  of  flow  through  the 
capillary  may  be  considerable,  a  correction  is  generally  necessary 
for  this  kinetic  energy,  which  is  transformed  into  heat.  Hagen- 
bach was  the  first  to  attempt  to  make  this  correction  but 
Neumann  (1858)  and  Jacobson  (1860)  were  the  first  to  put  the 
correction  into  satisfactory  form.  Thus  both  the  method  of 
measurement  and  the  formula  used  in  calculation  of  absolute  vis- 
cosities were  practically  the  same  by  1860  that  they  are  today. 
Unfortunately,  these  important  researches  have  not  been  suffi- 
ciently well-known,  hence  their  results  have  been  repeatedly 
rediscovered,  and  there  is  an  evident  confusion  in  the  minds  of 
many  as  to  the  conditions  necessary  for  exact  measurement. 
The  so-called  "transpiration"  or  Poiseuille  method  was  not  the 
only  one  which  was  worked  out  during  this  period  of  perfecting 
the  methods  of  measurement.  The  pendulum  method  was 
developed  by  Moritz  (1847),  Stokes  (1849),  O.  E.  Meyer  (1860), 
Helmholtz  (1860)  and  Maxwell  (1860).  The  well-known  method 
of  the  falling  sphere  was  worked  out  by  Stokes  (1849). 

During  the  period  to  which  we  have  just  referred,  Graham 
(1846-1862)  had  been  doing  his  important  work  on  gases,  but 
the  development  of  the  kinetic  theory  gave  a  great  impetus  to 
the  study  of  the  viscosity  of  gases;  and  at  the  hands  of  Maxwell, 
O.  E.  Meyer  and  others,  viscosity  in  turn  gave  the  most  striking 
confirmation  to  the  kinetic  theory.  The  work  on  the  viscosity 
of  gases  has  continued  on  until  the  present,  being  done  almost 
exclusively  by  physicists. 

To  chemists,  on  the  other  hand,  impressed  by  the  relations 
between  physical  properties  and  chemical  composition,  so  forcibly 
brought  to  their  attention  by  the  work  of  Kopp,  the  viscosity  of 
liquids  has  been  an  interesting  subject  of  study.  To  this  group 
belong  the  researches  of  Graham  (1861),  Rellstab  (1868),  Guerout 
(1875),  Pribram  and  Handl  (1878),  Gartenmeister  (1890), 
Thorpe  and  Rodger  (1893)  and  many  others. 


METHODS  OF  MEASUREMENT  3 

The  rise  of  modern  physical  chemistry  resulted  in  an  awaken- 
ing of  interest  in  all  of  the  properties  of  aqueous  solutions. 
Along  with  other  properties,  viscosity  received  attention  from  a 
great  number  of  physical  chemists,  among  whom  we  may  cite 
Arrhenius  (1887),  Wm.  Ostwald  (1893),  J.  Wagner  (1883-90), 
Reyher  (1888),  Miitzel  (1891).  It  must  be  admitted  that  our 
knowledge  of  viscosity  has  not  played  an  important  part  in  the 
development  of  modern  physical  chemistry.  It  is  doubtless  for 
this  reason  that  the  subject  of  viscosity  is  left  unconsidered  in 
most  textbooks  of  physical  chemistry.  It  is  certainly  not  be- 
cause viscosity  does  not  play  an  important  role  in  solutions,  but 
rather  that  the  variables  in  the  problem  have  not  been  properly 
estimated.  That  with  the  physical  chemist  viscosity  has  so 
long  remained  in  the  background,  makes  it  all  the  more  promis- 
ing as  a  subject  of  study,  particularly  since  it  is  becoming  more 
and  more  nearly  certain  that  viscosity  is  intimately  related  to 
many  very  diverse  properties  such  as  diffusion,  migration  of 
ions,  conductivity,  volume,  vapor-pressure,  rate  of  solution  and 
of  crystallization,  as  well  as  chemical  composition  and  consti- 
tution, including  association  and  hydration.  It  seems  probable 
that  the  work  in  this  field  is  going  to  expand  rapidly,  for  it  is 
becoming  imperative  that  the  exact  relation  between  viscosity 
and  conductivity,  for  example,  should  be  clearly  demonstrated. 

With  the  recent  advances  in  our  knowledge  of  the  nature  of 
colloids,  there  was  certain  to  be  an  extended  study  of  the  vis- 
cosity of  these  substances,  because  no  property  of  colloids  is  so 
significant  as  the  viscosity.  This  in  turn  has  again  stimulated 
interest  in  viscosity  on  the  part  of  the  physiologist,  so  that  the 
viscosity  of  blood,  milk,  and  other  body  fluids  have  been 
repeatedly  investigated  under  the  most  varied  conditions  during 
the  past  few  years. 

The  use  of  viscosity  measurements  for  testing  oils,  paints, 
and  various  substances  of  technical  interest  has  given  rise  to  a 
series  of  investigations,  that  of  Engler  (1885)  being  among  the 
earliest  and  most  important  in  this  group.  These  researches 
have  been  devoted  largely  to  devising  of  instruments  and  to  -a 
comparison  of  the  results  obtained. 

Quite  unrelated  to  the  above  groups  for  the  most  part,  are 
the  investigations  which  have  undertaken  to  study  the  viscosity 


4  FLUIDITY  AND  PLASTICITY 

of  solids.  The  study  of  elasticity  has  been  the  dominant  idea 
in  this  group  of  researches. 

Very  little  work  has  been  done  upon  the  viscosity  of  matter 
in  the  different  states  of  aggregation  taken  as  a  whole.  If  it 
has  been  shown  that  our  knowledge  of  viscosity  consists  of 
somewhat  unrelated  groups,  it  is  equally  apparent  that  such  a 
separation  is  artificial  and  that  nothing  could  be  more  important 
for  our  complete  understanding  of  viscosity,  than  to  bring  these 
groups  together  into  an  inter-related  whole.  We  shall  therefore 
not  make  an  attempt  to  follow  the  chronological  method,  where 
it  interferes  with  the  consideration  of  the  subject  as  a  whole. 
Nevertheless  the  groups  of  researches  to  which  we  have  alluded 
stand  out  rather  clearly.  The  methods  of  measurement  in  use 
will  be  first  considered,  after  which  we  shall  study  the  viscosities 
of  liquids,  solutions,  solids,  and  gases  respectively. 

Elastic  Deformation,  Plastic,  Viscous,  and  Turbulent  Flow. — 
If  a  perfectly  elastic  solid  be  subjected  to  a  shearing  stress  a 
certain  strain  is  developed  which  entirely  disappears  when  the 
stress  is  removed.  The  total  work  done  is  zero,  the  process  is 
reversible,  and  viscosity  can  play  no  part  in  the  movement. 
This  is  not  a  case  of  flow  but  of  elastic  deformation.  If  a  body 
which  is  imperfectly  elastic  as  regards  its  form  be  subjected  to 
shearing  stress,  it  will  be  found  that  a  part,  at  least,  of  the 
deformation  will  remain  long  after  the  stress  is  removed.  In  this 
case  work  has  been  done  in  overcoming  some  kind  of  internal 
friction.  We  may  distinguish  the  kinds  of  flow  under  three 
regimes.  It  is  characteristic  of  viscous  or  linear  flow  that  the 
amount  of  deformation  is  directly  proportional  to  the  deforming 
force,  and  the  ratio  of  the  latter  to  the  former  gives  a  measure 
of  viscosity.  It  has  been  questioned  at  times  whether  this  ratio 
is  truly  constant,  but  it  appears  that  only  one  qualification  is 
necessary.  In  very  viscous  substances  time  may  be  necessary 
for  the  flow  to  reach  a  steady  state,  aside  from  any  period  of 
acceleration,  because  with  substances  like  pitch  the  viscous 
resistance  develops  slowly,  so  that  the  above  ratio  gradually 
increases  when  the  load  is  first  put  on,  but  even  in  this  case  the 
ratio  finally  reaches  a  value  which  is  independent  of  the  amount 
of  the  load.  As,  however,  the  deforming  force  is  steadily  in- 
creased, a  point  may  be  reached  where  the  above  ratio  suddenly 


METHODS  OF  MEASUREMENT  5 

decreases.  At  this  point  the  regime  of  turbulent  or  hydraulic 
flow  begins.  This  will  be  studied  in  detail  at  a  later  point  in 
the  development  of  the  subject.  There  are  substances,  on  the 
other  hand,  for  which  the  value  of  the  above  ratio  increases 
indefinitely  as  soon  as  the  deforming  force  falls  below  a  certain 
minimum.  These  substances  are  said  to  be  plastic.  In  plastic 
flow  it  is  generally  understood  that  a  definite  shearing  force  is 
required  before  any  deformation  takes  place.  But  whether  this 
is  strictly  true  or  not  has  not  been  established. 

The  Coefficient  of  Viscosity. — Consider  two  parallel  planes  A 
and  B,  s  being  their  distance  apart.  If  a  shearing  force  F  per 
unit  area  give  the  plane  A  a  velocity  v  in  reference  to  B,  the 
velocity  of  each  stratum,  between  A  and  B,  as  was  first  pointed 
out  by  Newton,  will  be  proportional  to  its  distance  from  B. 
The  rate  of  shear  dv/ds  is  therefore  constant  throughout  a 
homogeneous  fluid  under  the  above  conditions.  The  possibility 
that  it  may  not  be  constant  near  a  boundary  surface  will  be 
considered  later.  Since  the  force  F  is  required  to  maintain  a 
uniform  velocity,  this  force  must  be  opposed  by  another  which 
is  equal  in  amount  due  to  the  internal  friction.  The  ratio  of 
this  force  to  the  rate  of  shear  is  called  the  coefficient  of  viscosity 
and  is  usually  denoted  by  the  symbol  17 

Fs 
*»-  (1) 

The  dimensions  of  viscosity  are  [ML-1T~1].  The  definition  of 
viscosity  due  to  Maxwell  may  be  stated  as  follows:  The  vis- 
cosity of  a  substance  is  measured  by  the  tangential  force  on  a 
unit  area  of  either  of  two  horizontal  planes  at  a  unit  distance  apart 
required  to  move  one  plane  with  unit  velocity  in  reference  to 
the  other  plane,  the  space  between  being  filled  with  the  viscous 
substance.  The  coefficient  of  fluidity  is  the  reciprocal  of  the 
coefficient  of  viscosity,  so  that  if  the  former  is  denoted  by  <f>  we 

have  0  =  -.     The  coefficient  of  fluidity  may  be  independently 

defined  as  the  velocity  given  to  either  of  two  horizontal  planes 
in  respect  to  the  other  by  a  unit  tangential  force  per  unit  area, 
when  the  planes  are  a  unit  distance  apart  and  the  space  between 
them  is  filled  with  the  viscous  substance. 


6  FLUIDITY  AND  PLASTICITY 

Methods  of  Measurement. — Almost  numberless  instruments 
have  been  devised  for  the  measurement  of  viscosity,  but  the 
greater  part  of  these  are  suitable  for  giving  relative  values  only. 
There  are,  however,  several  quite  distinct  methods  which  are 
susceptible  of  mathematical  treatment  so  that  absolute  viscosities 
may  be  obtained.  The  possible  methods  for  measuring  viscosity 
may  be  classified  under  three  heads  as  follows: 

1.  The  measurement  of  the  resistance  offered  to  a  moving 
body  (usually  a  solid)  in  contact  with  the  viscous  fluid. 

2.  The  measurement  of  the  rate  of  flow  of  a  viscous  fluid. 

3.  Methods  in  which  neither  the  flow  nor  the  resistance  to 
flow  are  measured. 

1.  The  various  methods  for  measuring  viscosity  while  maintaining  the 
fluid  in  a  nearly  fixed  position,  together  with  the  names  of  investigators 
who  have  developed  the  method  are  as  follows: 

(a)  A  horizontal  disk  supported  at  its  middle  point  by  a  wire  and  oscil- 
lating around  the  wire  as  an  axis.  Coulomb  (1801),  Moritz  (1847),  Stokes 
(1850),  Meyer  (1865),  Maxwell  (1866),  Grotrian  (1876),  Oberbeck  (1880), 
Th.  Schmidt  (1882),  Stables  and  Wilson  (1883),  Fawsitt  (1908). 

(6)  A  sphere  filled  with  liquid  and  oscillating  around  its  vertical  axis. 
Helmholtz  and  Piotrowski  (1868),  Ladenburg  (1908). 

(c)  A  cylinder  filled  with  liquid  and  oscillating  around  its  vertical  axis. 
Mutzel  (1891). 

(d)  Concentric  cylinders.     The  outside  one  is  rotated  at  constant  velocity 
and  the  torque,  exerted  upon  the  inner  coaxial  cylinder  which  is  immersed 
in  the  viscous  fluid,  is  measured.     Stokes  (1845),  de  St.   Venant  (1847), 
Boussinesq  (1877),  Couette  (1888),  Mallock  (1888),  Perry  (1893). 

(e)  An  oscillating  solid  sphere  immersed  in  the  viscous  substance  and 
supported  by  bifilar  suspension  was  used  by  Konig  (1885). 

(/)  A  body  moving  freely  under  the  action  of  gravity,  e.g.,  falling  sphere 
of  platinum,  mercury,  or  water,  a  falling  body  of  other  shape  than  a  sphere, 
a  rising  bubble  of  air.  Stokes  (1845),  Pisati  (1877),  Schottner  (1879),  de 
Keen  (1889),  O.  Jones  (1894),  Duff  (1896),  J.  Thomson  (1898),  Tammann 
(1898),  Schaum  (1899),  Allen  (1900),  Ladenburg  (1906),  Valenta  (1906), 
Arndt  (1907). 

2.  The  methods  for  measuring  the  rate  of  flow  of  a  viscous  fluid: 

(a)  Efflux  through  horizontal  tubes  of  small  diameter.  Gerstner  (1798), 
Girard  (1816),  Poiseuille  (1842),  G.  Wiedemann  (1856),  Rellstab  (1868), 
Sprung  (1876),  Rosencranz  (1877),  Grotrian  (1877),  Prlbram  and  Handl 
(1878),  Slotte  (1881),  Stephan  (1882),  Foussereau  (1885),  Couette  (1890), 
Bruckner  (1891),  Thorpe  and  Rodger  (1893),  Hosking  (1900),  Bingham  and 
White  (1912). 

(6)  Efflux  through  a  vertical  tube  of  small  diameter.     Stephan  (1882), 


METHODS  OF  MEASUREMENT  7 

Englcr  (1885),  Arrhenius  (1887),  Ostwald  (1893),  Gartenmeister  (1890), 
Heydweiller  (1895),  Friedlander  (1901),  Mclntosh  and  Steele  (1906), 
Rankine  (1910). 

(c)  Efflux  through  a  bent  capillary.     Griineisen  (1905). 

(d)  Bending  of  beams  and  torsion  of  rods  of  viscous  substance.     Trouton 
(1906),  Trouton  and  Andrews  (1904). 

(e)  Rate  at  which  one  substance  penetrates  another  under  the  influence 
of  capillary  action,  diffusion,  or  solution  tension. 

3.  Other  methods  for  measuring  viscosity: 

(a)  Decay  of  oscillations  of  a  liquid  in  U-shaped  tubes.     Lambert  (1784). 

(b)  Decay  of  waves  upon  a  free  surface.     Stokes  (1851),  Watson  (1902). 

(c)  Decay  of  vibrations  in  a  viscous  substance.     Guye  and  Mintz  (1908). 

(d)  Rate  of  crystallization.     Wilson  (1900). 

Nomenclature. — A  great  variety  of  names  have  been  given  to 
instruments  devised  for  measuring  viscosity,  among  which  we 
may  cite  viscometer,  viscosimeter,  glischrometer,  microrheom- 
eter,  stalagnometer,  and  viscostagnometer.  All  but  the  first 
two  are  but  little  used  and  their  introduction  seems  an  unneces- 
sary complication.  Viscometer  and  viscosimeter  are  about 
equally  used  in  England  and  America,  but  such  a  standard  work 
as  Watt's  Dictionary  uses  only  viscometer.  Viscosimeter  in  its 
German  equivalent  Viskosimeter  is  entirely  satisfactory,  but 
in  English  viscosimeter  is  apt  to  be  mispronounced  viscos- 
imeter. Furthermore  viscosimeter  does  not  so  easily  relate 
itself  in  one's  mind  to  viscometry  which  is  the  only  word  recog- 
nized in  the  standard  dictionaries  to  denote  the  measurement 
of  viscosity.  Professor  Brander  Matthews  kindly  informs  me 
that  the  formation  of  the  word  viscometer  is  quite  as  free  from 
objection  as  that  of  viscosimeter,  and  viscometer  is  in  harmony 
with  modern  spelling  reform.  Hence  viscometer  should  be 
adopted  as  the  name  for  all  instruments  used  for  measuring  vis- 
cosity. The  different  forms  are  distinguished  by  the  names  of 
their  inventors. 


CHAPTER  II 
THE  LAW  OF  POISEUILLE 

Experimental  Verification. — Prior  to  1842  it  had  not  been 
established  as  a  fact  that  the  movement  of  the  blood  through  the 
capillaries  has  its  origin  solely  in  the  contractions  of  the  heart. 
There  were  theories  current  that  the  capillaries  themselves 
caused  the  flow  of  blood  or  that  the  corpuscles  were  instrumental 
in  producing  it.  Poiseuille  reasoned  that  if  the  lengths  and 
diameters  of  the  capillaries  are  different  in  the  various  warm- 
blooded animals  and  if  the  pressure  and  temperature  of  the  blood 
vary  in  different  parts  of  the  body,  light  might  be  thrown  upon 
the  problem  by  investigating  the  effects  upon  the  rate  of  flow 
in  capillary  tubes  of  changes  in  (1)  pressure,  (2)  length  of  capil- 
lary, (3)  diameter  of  capillary,  and  (4)  temperature. 

The  results  of  Poiseuille's  experiments  were  of  a  more  funda- 
mental character  than  he  anticipated  for  they  proved  that  the 
conditions  of  capillary  flow  are  much  simpler  than  those  in  the 
wide  tubes  which  had  previously  been  employed,  and  by  his 
experiments  the  laws  of  viscous  flow  became  established.  Not 
only  did  Poiseuille  perform  experiments  which  resulted  in  the 
law  which  bears  his  name,  and  therefore  have  affected  all  subse- 
quent work,  but  he  measured  the  efflux  times  of  water  by  the 
absolute  method  taking  elaborate  precautions  to  insure  accuracy, 
and  using  capillaries  of  various  lengths  and  diameters  which  are 
equivalent  to  separate  instruments — in  all  over  forty  in  number. 
Thus  one  is  justified  in  studying  his  work  in  considerable  detail, 
not  only  for  its  historic  interest,  but  on  account  of  its  bearing 
upon  questions  which  will  arise  later.  In  the  Appendix  his 
measurements  are  reproduced  in  full. 

In  Fig.  1  is  shown  the  most  essential  part  of  the  apparatus  of 
Poiseuille.  It  consists  of  a  horizontal  glass  capillary  d  joined 
to  the  bulb,  whose  volume  between  the  marks  c  and  e  was  accu- 
rately determined.  The  bulb  is  connected  above  with  a  tube 
which  leads  to  (1)  a  60-1  reservoir  for  keeping  the  pressure  of  the 
air  within  the  apparatus  constant,  (2)  a  manometer,  filled  with 


THE  LAW  OF  PO I  SEVILLE  9 

water  or  mercury,  and  (3)  a  pump  which  is  used  for  giving  the 
desired  pressure.  The  capillary  opens  into  the  distilled  water 
of  the  bath  in  which  the  bulb  and  capillary  are  immersed.  After 
the  dimensions  of  the  bulb  and  capillary  have  been  found,  it  is 
only  necessary,  in  making  a  viscosity  determination  at  any  given 
temperature,  to  observe  the  time  necessary  for  a  volume  of 
liquid  equal  to  that  contained  in  the  bulb  to  flow  through  the 
capillary  under  a  determined  pressure.  Without  going  into  detail 
at  this  point,  it  need  be  merely  stated 
here  that  due  means  were  taken  for 
getting  the  true  dimensions  of  the 
capillary  and  bulb,  for  filling  the 
apparatus  with  clean  pure  liquid,  and 
for  estimating  the  mean  effective  pres- 
sure, which  consists  of  the  pressure 
obtained  from  the  manometer  plus 
the  hydrostatic  pressure  from  the  FlG-  1.— Poiseuiile's  viscome- 
bottom  of  the  falling  meniscus  in  the 

bulb  to  the  level  of  the  capillary,  minus  the  hydrostatic  pres- 
sure from  the  level  of  the  capillary  to  the  surface  of  the  bath, 
minus  a  correction  for  the  capillary  action  in  the  bulb,  and  two 
corrections  for  the  pressure  of  the  atmosphere,  which  may  be 
either  positive  or  negative.  One  of  these  last  corrections  is  due 
to  the  air  within  the  apparatus  being  more  dense  than  that 
outside,  the  other  is  due  to  the  difference  of  pressure  of  the  atmo- 
sphere upon  the  liquid  surfaces  in  the  upper  arm  of  the  manom- 
eter and  in  the  bath,  unless  they  happen  to  be  at  the  same  level. 
Law  of  Pressures. — In  obtaining  this  law  all  of  the  experi- 
ments were  made  at  a  temperature  of  10°C.  For  a  capillary  of 
given  length  and  diameter,  the  time  of  transpiration  was  meas- 
ured for  various  pressures.  For  example,  one  capillary  was  75.8 
mm  long,  the  major  and  minor  axes  of  the  end  of  the  capillary 
nearer  the  bulb  were  0.1405  and  0.1430  mm  and  those  of  the 
open  end  0.1400  and  0.1420  mm  respectively.  The  pressures 
used  are  given  in  the  first  column  of  Table  I  and  the  times  of 
transpiration  in  column  2.  One  of  these  values  is  then  employed 
to  calculate  the  others  on  the  assumption  that  the  times  of  tran- 
spiration are  inversely  proportional  to  the  pressures,  as  given  in 
column  3. 


10 


FLUIDITY  AND  PLASTICITY 
TABLE  I. — CAPILLARY  A' 


Pressure  in 

Observed  time 

millimeters  of 
mercury  at 
10°C 

for  transpiration 
of  13.  34085  cc 
of  water 

Calculated 
time 

Per  cent 
difference 

97.764 

10,361.0 

147  .  832 

6,851.0 

6,851.91 

0.01 

193  .  632 

5,233.0 

5,231.22 

0.03 

337.675 

2,612.5 

2,612.84 

0.01 

738.715 

1,372.5 

1,371.20 

0.09 

774  .  676 

1,308.0 

1,307.55 

0.04 

In  the  above  case  it  is  certainly  true  that  the  rate  of  flow  is 
proportional  to  the  pressure,  but  it  is  equally  certain  that  this 
relation  no  longer  holds  when  the  capillary  becomes  sufficiently 
shortened.  Thus  when  the  length  of  the  tube  used  above  is 
shortened  to  15.75  mm,  the  values  given  in  Table  II  are  obtained. 

TABLE  II. — CAPILLARY  A™ 


Pressure  in 

Observed  time 

millimeters  of        !     for  transpiration 

Calculated 

Per  cent 

mercury  at 

of  13  .  34085  cc 

time 

difference 

10°C 

of  water 

24.661 

8,646 

49.591 

4,355 

4,299 

-  1  .  29 

98.233 

2,194 

2,170 

-1.09 

148.233 

1,455 

1,438 

-1.17 

194.257 

1,116 

1,097 

-1.63 

388.000 

571 

549 

-3.85 

775.160 

298 

275 

-7.72 

Not  only  is  there  a  marked  deviation  from  the  assumed  law 
of  pressures  as  soon  as  the  capillary  is  sufficiently  shortened,  but 
the  percentage  difference  between  the  observed  and  calculated 
values  increases  quite  regularly  as  the  pressure  increases.  But 
in  either  case,  whether  the  capillary  is  shortened  or  the  pressure 
increased,  we  note  that  the  velocity  is  decreased.  Whether  the 
irregularity  here  observed  is  due  to  the  use  of  some  of  the  avail- 
able work  in  imparting  kinetic  energy  to  the  liquid,  or  it  is  due 


THE  LAW  OF  PO1  SEVILLE 


11 


to  eddy  currents  which  appear  under  conditions  of  hydraulic 
flow,  we  will  reserve  for  later  discussion.  This  question  was  not 
considered  by  Poiseuille,  yet  with  a  great  variety  of  tables  show- 
ing an  agreement  like  that  in  Table  I  above,  Poiseuille  was  fully 
justified  in  concluding  that  for  tubes  of  very  small  diameters 
and  of  sufficient  length,  the  quantity  of  liquid  which  transpires 
in  a  given  time  and  at  a  given  temperature  is  directly  proportional 
to  the  pressure,  or  V  =  Kp,  where  K  is  a  constant,  V  the  volume, 
and  p  the  pressure  head,  causing  the  flow  through  the  tube. 

Law  of  Lengths. — Poiseuille  next  studied  the  effect  of  the 
length  of  the  tube  upon  the  rate  of  flow,  but  this  problem  pre- 
sented exceptional  difficulty  owing  to  the  fact  that  tubes  are 
never  of  uniform  cross-section.  With  the  camera  lucida  he  ex- 
amined and  measured  each  section  of  the  tubes,  which  had  been 
carefully  selected  from  a  large  number,  and  finally  corrections 
were  made  for  the  small  changes  in  diameter,  assuming  the  law 
of  diameters  to  be  given  later.  This  seems  justified  since  the 
corrections  were  very  small.  In  Table  III  the  results  are  given 
which  Poiseuille  obtained  with  capillary  "B."  The  lengths  of 
the  capillary  are  given  in  column  1,  the  major  and  minor  axes 
of  the  free  end  in  column  2,  the  time  required  for  the  transpiration 

TABLE  III. — CAPILLARY  B 


Length  of 
tube  in 
millimeters 

Major  and 
minor  axes  of 
free  end 

Time  of 
transpiration  of 
6.4482  cc 

Time 
calculated 

Per  cent, 
difference 

100.050 

f  0.1135 
(0.1117 

2,052.98 

75.050 

0.1140 
0.1120 

1,526.20 

1,539.0 

0.85 

49.375 

0.1142 
0.1122 

998.74         !   1,004.0 

0.53 

23  .  575 

0.1145\ 
\  0.1123] 

475  .  18 

476.8 

0.34 

9.000 
3.900 

f  0.1144\ 
J0.1124/ 
j  0.1145  \ 
\  0.1125/ 

199.39 
110.20 

181.4 
86.4 

-9.05 
-21.64 

12 


FLUIDITY  AND  PLASTICITY 


of  the  6.4482  cc  of  water  at  10°C  contained  in  the  bulb  at  a 
constant  pressure  of  775  mm  of  mercury  are  given  in  column  3. 
Assuming  that  the  time  of  flow  is  directly  proportional  to  the 
length  of  the  tube,  Poiseuille  used  the  time  of  one  experiment 
to  calculate  the  one  immediately  succeeding,  and  thus  are  ob- 
tained the  values  given  in  column  4.  It  is  evident  that  the  last 
two  lengths  are  too  short,  but  the  others  fairly  substantiate  the 
law.  The  agreement  is  still  better  when  corrections  are  made 
for  the  varying  diameters  of  the  tube.  This  correction  is  espe- 
cially important  since,  as  will  be  shown,  the  efflux  rate  varies 
as  the  fourth  power  of  the  diameter.  From  results  like  those 
exhibited  in  Table  III  Poiseuille  concluded  that  the  quantity  of 
liquid  passing  through  a  tube  of  very  small  diameter  at  a  given 
temperature  and  pressure  varies  inversely  as  the  length,  and  we  have 
that  V  =  K"p/l  where  I  represents  the  length.  But  the  last 
two  observations  show  that  this  law  has  its  limitations. 

Law  of  Diameters. — To  discover  the  relation  between  the 
diameter  of  the  capillary  and  the  rate  of  flow,  Poiseuille  calculated 
the  quantity  of  water  which  would  flow  through  25  mm  of  the 
different  tubes  at  10°C  under  a  pressure  of  775  mm  of  mercury 
in  500  seconds,  obtaining  the  values  given  in  Table  IV. 

TABLE  IV 


Designation 
of  tube 

Mean  diameter  ' 
of  tube  in 
centimeters 

Volume  efflux 
in  500  sec.  from 
observations 

Volume 
calculated 

'    Per  cent, 
difference 

M                 0.0013949 

0.0014648 

0.001465 

+0.02 

E                   0.0029380 

0.0288260  ! 

0.028808 

-0.07 

D 

0.0043738 

0.1415002  : 

0.141630 

+0.10 

C 

0.0085492 

2.0673912  i 

2.066930 

-0.02 

B                  0.0113400 

6.3982933  ! 

6.389240 

-0.14 

A                  0.0141600 

15.5328451  j 

15.547100 

+0.10 

F 

0.0652170 

6,995.8702463 

The  volumes  calculated  in  the  fourth  column  are  obtained  by 
comparing  each  tube  with  the  one  following  on  the  assumption 
that  the  quantity  traversing  the  tube  is  proportional  to  the  fourth 
power  of  the  diameter,  thus  0.0029384 :  0.00139494  =  0.028826:  x, 
or  x  =  0.001465.  The  agreement  is  very  satisfactory,  hence  the 


THE  LAW  OF  POISE UILLE  13 

pd* 
formula  becomes  V  =  K  ~j-  •     For  water  at  10°C  he  found  the 

value  of  K  to  be  quite  exactly  2,495,224,  p  being  expressed  in 
millimeters  of  mercury  at  10°  and  I  and  d  in  centimeters.  He 
experimented  with  alcohol  and  mixtures  of  alcohol  and  water 
and  for  these  we  obtain  different  values  of  K.  Poiseuille  did 
not  use  the  terms  viscosity  or  fluidity,  nevertheless  these  values 
of  K  are  proportional  to  the  fluidity. 

The  Effect  of  Temperature  on  the  Rate  of  Flow. — Girard  had 
given  a  formula  to  represent  the  flow  of  water  in  a  pipe  as  a 
function  of  the  temperature,  but  the  constants  had  to  be  deter- 
mined for  each  pipe.  Poiseuille  gave  a  formula  for  capillary 
tubes  which  was  independent  of  the  instrument  used, 

Q  =  1,836,724,000(1  +  0.033679377  +  0.0002209936772)^ 

where  Q  represents  the  weight  of  water  traversing  the  capillary 
in  a  unit  of  time.  The  adequacy  of  this  formula  to  reproduce 
the  observed  values  is  shown  in  Table  V. 

TABLE  V.— CAPILLARY  A 

I  =  10.05    cm     d  =  0.0141125  cm     p  =  776    mm    of  mercury.     Time   of 
flow  1,000  sec. 

WEIGHT  OF  EFFLUX 

WEIGHT  OF  EFFLUX  CALCULATED  BY 

TEMPERATURE  OBSERVED  FORMULA 

0.6  5.74376  5.73955 

5.0  6.60962  6.60381 

10.0  7.64649  7.64435 

15.0  8.74996  8.74705 

20.0  9.91530  9.91191 

25.0  11.14584  11.13892 

30.1  12.45631  12.45423 
35.1  13.80695  13.80710 
40.1  15.21866  15.22184 
45.0  16.67396  16.66860 

Since  the  values  calculated  are  weights  and  not  volumes,  the 
values  of  Q  are  not  proportional  to  the  fluidity.  This  formula 

pd* 
remains  empirical,  but  the  expression  V  =  K  — ,—  can  be  readily 

derived  from  the  fundamental  laws  of  motion. 

Theoretical  Derivation  of  the  Law. — Hagenbach  (1860)  appears 
to  have  been  the  first  to  give  a  definition  of  viscosity.  He  made 


14  FLUIDITY  AND  PLASTICITY 

a  very  careful  study  of  the  earlier  work  on  viscosity  and  gave  a 
theoretical  derivation  of  the  law  of  Poiseuille,  which  has  had 
very  great  effect  upon  the  succeeding  history  of  this  subject. 
Neumann  gave  the  deduction  of  the  Law  of  Poiseuille  in  his 
lectures  on  Hydrodynamics  in  1858,  and  thus  prior  to  the  publi- 
cation of  Hagenbach's  paper  in  March,  1860.  This  deduction 
was  first  published  by  Jacobson  early  in  1860  and  the  lectures 
were  published  in  full  in  1883.  In  April,  1860  Helmholtz  pub- 
lished the  derivation  of  the  law  from  the  equations  of  motion. 
J.  Stephan  (1862)  and  Mathieu  (1863)  gave  independent  deriva- 
tions of  the  law.  Reference  should  also  be  made  to  the  treat- 
ment of  the  flow  in  long  narrow  tubes  by  Stokes  (1849). 

Imagine  a  horizontal  capillary  whose  bore  is  a  true  cylinder  to 
connect  two  reservoirs  L  (left)  and  R  (right)  there  being  a  differ- 
ence of  pressure  between  the  two  reservoirs,  at  the  level  of  the 
capillary,  amounting  to  p  grams  per  square  centimeter.  If  the  pres- 
sure in  L  is  the  greater  the  direction  of  flow  through  the  capillary 
will  be  from  left  to  right.  The  total  effective  pressure  p  is  used 
up  in  doing  various  forms  of  work,  several  of  which  can  be  differ- 
entiated with  a  resultant  gain  in  clearness  of  understanding  of 
the  conditions  of  flow. 

1.  Near  the  entrance  to  the  capillary  the  particles  of  fluid 
undergo  a  rapid  acceleration;  this  absorption  of  kinetic  energy 
causes  a  fall  in  the  pressure  amounting  to  pk. 

2.  Within  the  capillary,  there  may  be  a  finite  movement  of 
the  fluid  over  the  walls  of  the  tube,  due  to  slipping.     Unless 
the  external  friction  is  zero  or  infinity,  work  will  be  done  and 
there  will  be  a  fall  of  pressure  ps. 

3.  Unless  the  external  friction  is  zero,   the  layers  of  fluid 
nearer  the  walls  of  the  tube  will  move  more  slowly  than  the 
layers  nearer  the  axis  of  the  tube,  and  an  absorption  of  pressure 
due  to  this  internal  friction  will  result.     Let  this  be  pv. 

4.  If  the  path  of  the  particles  through  the  capillary  is  not 
perfectly  linear,  the  additional  distance  travelled  in  the  eddies, 
will  give  rise  to  a  further  drop  in  the  pressure  amounting  to  pe. 
This  turbulent  flow  is  certain  to  occur  when  the  velocity  of  flow 
becomes  sufficiently  high. 

5.  But  even  before  the  velocity  becomes  turbulent  it  seems 
possible  that  the  stream  lines  at  the  extremities  of  the  tube  may 


THE  LAW  OF  POI SEVILLE  15 

be  somewhat  distorted,  in  which  case  there  must  be  a  drop  in 
pressure  pe. 

6.  Heat  is  produced  as  the  fluid  passes  through  the  tube  and 
therefore  the  temperature  may  be  different  at  different  points  of 
the  tube  and  since  the  temperature  greatly  affects  the  viscosity 
of  most  substances,  this  may  affect  the  amount  of  work  done  in 
the  passage  through  the  tube.  If  the  fluid  is  incompressible  it 
will  have  the  same  mean  velocity  through  each  cross-section  of 
the  capillary,  and  the  pressure  must  fall  in  a  linear  manner  at  least 
so  long  as  the  flow  is  linear.  If  on  the  other  hand  the  substance 
is  compressible,  the  velocity  must  increase  as  the  fluid  passes 
through  the  tube,  because  of  the  expansion  which  results  from 
the  decrease  of  pressure.  With  the  expansion  there  is  a  lowering 
of  the  temperature.  Let  the  resultant  effect  of  these  changes  in 
the  temperature  upon  the  effective  pressure  be  pT.  It  may  be 
either  positive  or  negative. 

At  the  exit  of  the  capillary  the  fluid  has  no  effective  pressure 
but  it  still  possesses  all  of  its  kinetic  energy  which  causes  the 
fluid  to  go  for  a  considerable  distance  out  into  the  reservoir  R, 
dragging  some  of  the  fluid  in  R  with  it  and  producing  eddies,  so 
that  the  kinetic  energy  is  finally  dissipated  in  overcoming  viscous 
resistance  outside  of  the  capillary,  and  not  in  adding  to  the  effec- 
tive pressure,  as  Applebey  (1910)  has  supposed. 

The  sum  of  these  possible  losses  of  effective  pressure  is  then 

p  =  pk  +  ps  +  Pv  +  pe  +  ps  +  PT  (2) 

We  shall  consider  first  the  case  where  p  =  pv,  supposing  that  the 
fluid  is  incompressible,  as  is  nearly  the  case  in  liquids. 

Let  the  radius  of  the  capillary  be  R  and  the  radius  of  a  hollow 
cylinder  coaxial  with  the  capillary  be  r.  It  is  evident  from  the 
symmetrical  arrangement  that  at  every  point  in  such  a  cylinder, 
the  velocity  must  be  identical.  Let  this  velocity  be  v.  The 

dv 
rate  of  deformation  must  be  -r;  and  the  tangential  force  due  to 

dv 
the  viscous  resistance,  acting  from  right  to  left,  will  be  77  -3-  (cf. 

Eq.  (1)).     Over  the  whole  surface  of  the  cylinder  whose  length 
is  /,  this  force  must  amount  to 

dv 


16  FLUIDITY  AND  PLASTICITY 

But  the  force  due  to  the  frictional  resistance  on  the  outside  of 
the  cylinder  must  be  exactly  balanced  by  a  force  due  to  the  pres- 
sure and  this  is 

-  irr2pg 

where  p  is  the  pressure  in  grams  per  square  centimeter  and  g  is 
the  acceleration  due  to  gravity.  The  negative  sign  is  used 
because  this  force  acts  from  left  to  right.  We  have  then  that 


but  v  =  0  when  r  =  R,  therefore  the  constant  of  integration  K 
can  be  evaluated 

JT         ™R* 
/Y    —       Aj 

4/77 
,-J*(«.-0  (3) 

From  Eq.  (3)  we  may  obtain  the  velocity  in  centimeters  per 
second  at  any  point  in  the  capillary.  It  follows  that  the  liquid 
flowing  through  the  capillary  in  a  given  time  has  the  volume  of 
a  paraboloid  of  revolution.  If  the  volume  per  second  is  U,  then 


=  0 

which  is  the  Law  of  Poiseuille.     If  V  is  the  total  volume  of  efflux 
in  the  time  t,  the  formula  becomes 


r  -  (v 

Sir, 

The  mean  velocity  of  the  fluid,  in  cubic  centimeters  per  second 
passing  through  the  tube,  /,  is 

'-^-f 

Summary. — The  simple  law  of  Poiseuille  was  first  discovered 
experimentally,  after  which  its  theoretical  deduction  was  quickly 
made.  There  is,  however,  a  considerable  amount  of  data  for 
which  the  simple  law  is  not  sufficient.  The  law  may  be  given 
far  greater  usefulness  by  adding  certain  correction  terms,  which 
are  the  subject  of  discussion  in  the  following  chapter. 


CHAPTER  III 
THE  AMPLIFICATION  OF  THE  LAW  OF  POISEUILLE 

The  Kinetic  Energy  Correction. — In  deriving  the  law  in  the 
preceding  chapter,  we  limited  ourselves  to  the  simplest  case, 
where  all  of  the  energy  is  employed  in  overcoming  viscous  resis- 
tance within  the  fluid,  or  p  =  pv.  It  is  desirable  however  that 
the  law  be  given  a  wider  application,  and  that  the  law  be  tested 
under  the  most  varied  conditions.  In  the  experiments  which 
Poiseuille  used  to  verify  his  law,  the  kinetic  energy  correction  was 
negligible,  but  the  time  necessary  for  a  single  determination  was 
often  excessive,  consuming  several  hours.  It  is  to  be  recalled  at 
this  point  that  in  some  of  his  experiments,  in  which  the  rate  of  flow 
was  higher  than  in  the  others,  the  law  was  not  verified.  Poise- 
uille and  others  have  been  greatly  troubled  in  their  viscosity  de- 
terminations by  dust  particles  becoming  lodged  in  the  capillary. 
If  it  were  possible  therefore  to  employ  higher  speeds,  not  only 
would  there  be  an  economy  in  time  but  the  dust  particles  would 
be  much'  more  likely  to  be  swept  out  from  the  tube.  However 
in  using  these  higher  velocities  a  correction  for  the  loss  in  kinetic 
energy  must  be  applied. 

Hagenbach  (1860)  is  the  first  one  to  attempt  to  make  this 
correction,  the  results  of  whose  work  became  generally  known, 
although  it  appears  that  Neumann  prior  to  1860  had  made  the 
correction  in  nearly  its  present  form.  The  work  of  Neumann 
was  reported  by  Jacobson  in  1860  but  his  work  has  also  remained 
but  little  known  to  workers  in  this  field.  Gartenmeister  (1890) 
reported  that  Finkener  had  arrived  at  a  correction  which  differed 
from  that  of  Hagenbach,  but  Finkener  seems  not  to  have  pub- 
lished any  monograph  on  the  subject  stating  why  he  considered 
his  correction  superior.  However  Couette  in  the  same  year 
(1890)  published  a  very  important  paper  in  which  he  arrived 
independently  at  the  same  correction  as  that  given  by  Neumann 
and  Finkener,  and  a  year  later  Wilberforce  (1891)  independently 
attacked  the  same  subject  and  showed  that  there  is  a  slip  in  the 
2  17 


18  FLUIDITY  AND  PLASTICITY 

reasoning  of  Hagenbach.  He  showed  that  Hagenbach  should 
have  reached  a  value  which  is  identical  with  that  given  by  the 
others.  The  correction  may  be  simply  deduced  as  follows: 

The  kinetic  energy  of  the  fluid  passing  any  cross-section  of  a 
cylindrical  tube  per  unit  of  time  is 


where  p  is  the  density  of  the  fluid.  Since  the  volume  of  fluid 
passing  any  cross-section  per  unit  of  time  is  TrR2I,  the  energy  sup- 
plied in  producing  the  flow  is  irR2Ipg,  hence,  the  energy  converted 
into  heat  within  the  tube  must  be  irR2I(pg  —  p/2).  From  Eqs.  (2) 
and  (6)  we  have 


P) 

Thus  taking  into  account  the  loss  in  kinetic  energy,  the  formula 
of  Poiseuille  becomes 

mpV 


SVl  ftrft 

in  which  m  is  a  constant  which  according  to  the  above  derivation 
is  equal  to  unity.  The  formula  of  Hagenbach  differed  only  in 
that  the  constant  m  is  equal  to  2  ~^  or  0.7938. 

It  is  of  historical  interest  in  this  connection  to  note  that  Ber- 
nouilli's  assumption  that  all  of  the  particles  flowing  through  a  pipe 
have  the  same  velocity,  leads  one  to  the  conclusion  that  the 
kinetic  energy  of  the  fluid  passing  any  cross-section  per  unit  of 

irR2!3 
time  is  exactly  one-half  of  that  given  above  or  —  ^  —  '  an<^  the 

value  of  m  in  that  case  would  be  only  0.50.  This  value  was  actu- 
ally suggested  by  Reynolds  (1883)  when  the  openings  of  the  tubes 
were  rounded  or  trumpet-shaped,  but  m  =  0.752  when  the  ends 
are  cylindrical.  It  may  be  added  that  Hagenbach  compared  his 
value  of  0.7938  with  the  observed  values  obtained  by  various 
hydraulicians  working  with  wide  tubes,  Hagen  0.76,  Weisbach 
0.815,  Zeuner  0.80885,  Morin  0.82,  and  Bossut  0.807,  and  he 
found  that  his  value  was  near  the  mean.  But  account  should 
have  been  taken  of  the  fact  that  their  results  apply  to  the  tur- 
bulent regime,  but  not  necessarily  to  the  regime  of  linear  flow. 
Boussinesq  (1891)  while  admitting  the  correctness  of  the 


AMPLIFICATION  OF  THE  LAW  OF  POI SEVILLE 


19 


method  used  by  Couette — and  as  we  have  seen,  also  by  Neumann, 
Finkener,  and  Wilberforce — as  a  first  approximation,  gives  a 
more  rigorous  treatment  of  the  subject  on  the  basis  of  the  kinetic 
theory  by  which  he  finds  m  =  1.12. 

Knibbs  (1895)  in  a  valuable  discussion  of  the  viscosity  of  water 
by  the  efflux  method  has  studied  carefully  the  data  of  Poiseuille 
and  Jacobson  in  the  effort  to  find  the  value  of  m  which  would  most 
nearly  accord  with  the  experimental  results.  Throwing  Eq.  (8) 
in  the  form 

8?;  VI       mpV2  /r.N 

Pi      =      fT^        1 O       T1AA    '  (9) 

TfgK        ir^gK  t 
we  observe  that  since  for  a  given  tube  and  liquid  only  p  and  t 


15 

^ 

^ 

• 

14 

^ 

k>      13 

^ 

*" 

tf 

% 

q  n 

Q.  



o—  • 

.  ' 

3& 

g 

r*B 

>n 

c^.J 
x    g 

-cT- 

^ 

-t-    ° 
^i 

jy 

*^ 

^  — 

^ 

*°" 

s 

^ 

J 

S1 

1 

•)   11 

FIG.  2. — Finding  the  value  of  m  for  the  kinetic  energy  correction. 


vary,  this  is  the  equation  of  a  straight  line  and  may  be  written, 


(9a) 


where  a  and  6  are  constants.  Plotting  the  values  of  1/t  as  abscis- 
sas and  of  pt  as  ordinates  Knibbs  obtained  the  curves  shown  in 
Fig.  2,  using  the  data  for  Poiseuille's  tubes  Av,  Avn,  Bv,  and  Cv. 
When  t  becomes  very  great  the  corrective  term  vanishes  and  pt 
=  a.  The  values  of  a  are  given  by  the  intercepts  of  the  curves 
with  the  axis  of  ordinates.  The  tangent  of  the  angle  which  a 
line  makes  with  the  axis  of  abscissas  gives  the  value  of  b,  from 
which  the  value  of  m  is  obtained,  since 


20 


FLUIDITY  AND  PLASTICITY 


Using  a  combination  of  numerical  and  graphical  methods  the 
following  values  were  obtained. 

TABLE  VI. — VALUES  OF  ra  DEDUCED  BY  KNIBBS  FROM  POISEUILLE'S 
EXPERIMENTS 


Tube 

Length  in  centi- 
meters 

Mean  radius  in 
centimeters 

Values  of  m 

A111  

2  55 

0  00708 

04 

A™  

1  57 

0.00708 

.02 

Av 

0  95 

0  00708 

15 

AVI 

0  68 

0  00708 

08 

Avn 

0  10 

0  00708 

12 

B  
BIV  
Bv  
Cv  
F1  

10.00 
0.90 
0.39 
0.60 
20.00 

0.00567 
0.00567 
0.00567 
0.00427 
0.03267 

.23 
.14 
.03 
.87* 
.08 

F" 

9  97 

0  03267 

33 

F111 
F™  
Fv  

5.04 
2.60 
1.07 

0.03267 
0.03267 
0.03267 

1.16 
0.82* 

0.82* 

The  mean  is  1.14  or  rejecting  the  values  for  Cv,  FIV,  and  Fv, 
1.13.  Certain  of  the  tubes,  viz.,  A,  A1,  A11,  B1,  B",  B111,  C, 
C1,  C",  C111,  CIV,  D,  D1,  Du,  Dni,  DIV,  E,  E1,  E",  and  F  give 
no  satisfactory  indication  of  the  value  of  m.  Knibbs  deduced 
the  value  of  m  from  34  series  of  experiments  made  by  Jacobson 
and  obtained  an  average  value  of  1 . 14.  This  seems  like  a  remark- 
able justification  of  the  deduction  of  Boussinesq.  But  it  should 
be  added  that  the  individual  values  vary  from  0.82  to  1.44,  yet 
perhaps  this  variation  in  the  values  of  m  should  not  be  over- 
emphasized since  in  some  instances  the  amounts  of  the  corrections 
are  much  smaller  than  the  discrepancies  among  the  observa- 
tions themselves.  Knibbs  thinks  that  the  values  do  vary  more 
than  can  possibly  be  accounted  for  by  the  experimental  error  and 
that  possibly  the  value  of  m  is  not  a  constant  for  all  instruments. 
It  is  highly  desirable  that  further  experiments  be  undertaken  to 
determine  whether  m  is  a  constant  and  equal  to  1.1 2  or  if  it  is 
not  constant,  the  manner  of  its  variation. 


AMPLIFICATION  OF  THE  LAW  OF  POI SEVILLE 


21 


To  the  present  writer  it  seems  probable  that  the  kinetic  energy 
correction  is  truly  constant  for  all  tubes  which  are  perfect  cylin- 
ders. Irregularities  in  the  bore  of  the  tubes  will,  however,  have 
very  great  influence  in  altering  the  amount  of  the  correction, 
since  the  correction,  cf.  Equation  (7),  depends  upon  the  fourth 
power  of  the  radius  of  the  tube.  The  shape  of  the  ends  of  the 
capillary  has  already  been  referred  to  in  this  connection,  but  it 
seems  preferable  to  consider  the  effect  of  the  shape  of  the  ends 
of  the  tube  as  quite  distinct  from  the  kinetic  energy  correction. 
There  has  been  a  tendency  among  many  recent  experimenters 
to  overlook  the  kinetic  energy  correction  altogether,  which  is 
quite  unjustifiable.  We  have  indicated  that  it  is  not  practicable 
to  make  the  correction  negligible.  The  only  course  open  seems 
therefore  to  be  to  select  a  capillary  which  has  as  nearly  as  possi- 
ble a  uniform  cylindrical  (or  elliptical)  cross-section,  to  assume 
that  m  for  such  a  tube  has  the  constant  value  of  1.12,  but  to 
arrange  the  conditions  of  each  experiment  so  that  the  kinetic 
energy  correction  will  not  exceed  1  or  2  per  cent  of  the  viscosity 
being  measured.  In  this  case  an  error  of  several  per  cent  in  the 
value  of  the  constant  will  not  affect  the  result,  unless  an  accuracy 
is  desired  which  is  higher  than  has  yet  been  attained.  If  such  an 
accuracy  is  desired  the  value  of  m  should  be  found  for  each  tube 
by  the  method  of  Knibbs  which  has  been  discussed  above,  or  by 
the  method  employed  by  Bingham  and  White  (1912),  which  will 
be  described  below  in  dis- 
cussing the  alteration  in  the 
lines  of  flow  at  the  ends  of 
the  tube. 

Correction  for  Phenomena 
of  the  Flow  Peculiar  to  the 
Ends  of  the  Tube. — If  two 
tubes  of  large  diameter  are 
connected  by  a  short  capillary,  the  lines  of  flow  will  be  as 
represented  in  Fig.  3,  the  direction  of  flow  being  readily  visible 
in  emulsions,  suspensions,  or  when  a  strongly  colored  liquid  is 
allowed  to  flow  out  from  a  fine  tube  in  the  body  of  colorless 
liquid  near  the  entrance  to  the  capillary,  as  was  done  by 
Reynolds  (1883).  In  the  reservoir  at  the  entrance  A  there  is 
apparently  no  disturbance  until  the  opening  of  the  capillary  is 


FIG.    3. — Diagr 


22  FLUIDITY  AND  PLASTICITY 

almost  reached,  and  there  the  acceleration  is  very  rapid.  Even 
when  the  stream  lines  in  the  main  part  of  the  capillary  are  linear, 
it  seems  theoretically  necessary  to  assume  that  there  is  a  choking 
together  of  the  stream  lines  near  the  entrance  as  indicated  at  c. 
It  has  been  suggested  that  this  effect  might  be  prevented  by 
using  rounded  or  trumpet-shaped  openings  as  indicated  at  d. 

At  the  exit  of  the  capillary,  the  stream  continues  on  into 
the  reservoir  B  for  a  considerable  distance  with  its  diameter 
apparently  unchanged.  However  the  fall  in  pressure  of  the 
liquid  passing  through  the  large  tube  B  is  negligible,  so  that  the 
flow  observed  just  beyond  the  exit  takes  place  at  the  expense — 
not  of  pressure — but  of  kinetic  energy  taken  up  at  the  entrance. 
There  is  no  distortion  of  the  stream  lines  just  within  the  exit  end 
of  the  capillary,  and  it  is  not  clear  that  any  correction  at  this  end 
is  necessary,  under  the  conditions  which  we  have  depicted.  If 
the  capillary  opens  into  the  air,  there  will  naturally  be  a  capil- 
larity correction  and  the  shape  and  material  of  the  end  of  the 
tube  will  be  of  importance — cf.  Ronceray  (1911). 

That  the  stream  should  continue  for  some  distance  beyond 
the  exit  with  apparently  constant  diameter  seems  at  first  sight 
quite  surprising,  as  one  might  suppose  that  the  stream  would  at 
once  drag  along  the  adjacent  fluid.  The  explanation  is  not  far  to 
seek.  In  the  first  place  one  should  remember  that  the  velocities 
even  in  the  capillary  are  by  no  means  uniform.  Equation  (3) 
tells  us  that  particles  which  at  a  given  moment  are  in  a  plane 
surface  mno  will  after  a  certain  time  has  elapsed  be  in  a  paraboloid 
surface  mpo.  The  transition  from  the  stationary  cylinder  of 
fluid  in  contact  with  the  wall  to  the  coaxial  cylinders  having 
high  speed  is  apparently  abrupt.  As  the  exit  of  the  capillary  is 
passed,  there  is  nothing  to  prevent  the  larger  mass  of  liquid 
from  being  drawn  along  except  its  own  inertia.  But  the  rate  at 
which  the  kinetic  energy  of  the  inner  coaxial  cylinders  of  fluid 
passes  out  into  the  outer  cylinders  is  proportional  to  the  viscosity 
of  the  medium  and  to  the  area  of  the  cylinder.  Thus  in  a  fluid  of 
low  viscosity  a  capillary  stream  will  penetrate  for  some  distance. 
The  stream  disappears  rather  suddenly  due  probably  to  the 
development  of  eddies. 

Couette  has  attempted  to  evaluate  the  effects  of  the  ends  of 
the  tubes  by  supposing  that  they  are  equivalent  to  an  addition  to 


AMPLIFICATION  OF  THE  LAW  OF  PO I  SEVILLE  23 

the  actual  length  of  the  capillary,  which  he  represents  by  A.  The 
corrected  viscosity  7?c  should  therefore  be  calculated  by  the 
formula 

_    _     irgpRH  mpV_  (1Q) 


According  to  Couette  the  corrected  viscosity  is  always  a  little 
smaller  than  that  calculated  by  means  of  Eq.  (8)  and  we  obtain 
the  relation 

77    _  I  +A 

r,c  I 


Since  A  may  be  presumed  to  be  the  same  for  tubes  of  equal 
diameter  but  of  unequal  lengths  I  and  /',  one  should  obtain 
different  viscosities  17  and  77'  by  applying  Eq.  (8)  to  the  same  fluid. 
There  would  thus  be  the  relation 


Tic 


(11) 


To  test  out  his  theory,  Couette  used  experimental  results  of 
Poise  uille  with  tubes  A™  and  Av  which  gave  poor  agreement  with 
the  simple  law,  Eq.  (5)  cf.  Table  II,  VII  and  VIII.  The  efflux 
times  are  given  in  column  1,  the  viscosities  rjp  calculated  from 
the  simple  Poiseuille  formula  (5),  in  column  2,  the  more 
nearly  correct  viscosities  r?  and  rj',  calculated  from  Eq.  (8)  taking 
m  =  1.00,  in  column  3. 

TABLE    VII.  —  VISCOSITY    OF    WATER    CALCULATED    FROM    POISEUILLE'S 

EXPERIMENTS  WITH  TUBE  A" 
For  dimensions  cf.  Appendix  D,  Table  I,  p.  331 


Time 


r,P  Eq   (5) 


77  Eq.  (8),  m  =  1.00 


8.646 

0.01332 

0.01328 

4,355 

0.01349 

0.01339 

2,194 

0.01347 

0.01332 

1,455 

0.01347 

0.01324 

1,116 

0.01355 

0.01325 

571 

0.01384 

0.01325 

298 

0.01443 

0.01330 

24 


FLUIDITY  AND  PLASTICITY 


TABLE    VIII. — VISCOSITY    OF    WATER    CALCULATED    FROM    POISEUILLE'S 

EXPERIMENTS  WITH  TUBE  A" 
For  dimensions  cf.  Appendix  D,  Table  I,  p.  331 


Time 


.  (5) 


,'  Eq.  (8),  m  =  1.00 


3,829 

0.01383 

0.01363 

1,924 

0.01404 

0.01363 

994 

0.01442 

0.01363 

682 

0.01479 

0.01364 

537 

0  01512 

0.01366 

291 

0.01651 

0.01382 

165 

0.01863 

0.01388 

The  values  of  77  vary  but  little  around  the  mean  0.01329,  while 
the  values  of  t\v  show  a  regular  progression,  thus  demonstrating 
the  importance  of  the  kinetic  energy  correction.  The  first  three 
values  of  77'  in  Table  VIII  are  constant  and  equal  to  0.01363. 
The  last  four  values  show  a  steady  increase  which  may  be  due  to 
turbulent  flow  at  such  high  velocities.  From  17  and  77',  which  are 
notably  different  in  value,  the  corrected  viscosity  77C  as  well  as  the 
value  of  A  may  be  obtained  by  the  use  of  Eq.  (11).  We  get 
77C  =  0.01303  and  A  =  0.041  cm.  The  mean  diameter  of  these 
tubes  was  0.01417  cm  hence,  the  fictitious  elongation  of  the  tube 

is  a  little  less  than   three  times  the   diameter  i^-p  =  2.868)  • 

Couette  also  obtained  the  corrected  viscosity  directly  by 
experiment,  in  a  very  ingenious  manner.  He  employed  two 
capillaries  simultaneously,  which  had  the  same  diameter  but 
different  lengths.  The  arrangement  of  his  apparatus  is  shown  in 
Fig.  4,  where  T\  and  772  are  the  two  capillaries  connecting  three 
reservoirs  M,  N,  and  P.  The  pressure  in  each  reservoir  is 
measured  on  the  differential  manometer  H.  Since  the  volume  of 
efflux  through  both  capillaries  is  the  same  and  may  be  calculated 
from  the  increase  in  weight  of  the  liquid  in  the  receiving  flask  D, 
we  obtain  from  Eqs.  (7)  and  (9)  the  relation 

ZLJB*). 


V 


or 


+  A)          8Vr,c(l2  -A) 


i  -  pz 


r]c         - 


AMPLIFICATION  OF  THE  LAW  OF  PO I  SEVILLE 


25 


By  thus  eliminating  the  correction  for  the  kinetic  energy  and  the 
ends  of  the  tubes,  Couette  obtained,  for  the  corrected  viscosity 
(TJC)  of  water  at  10°,  0.01309  which  is  in  excellent  agreement  with 
the  value  calculated  above  from  Poiseuille's  experiments.  If, 
on  the  other  hand,  the  viscosity  (77)  is  calculated  by  means  of 
Eq.  (8)  with  m  =  1.00  for  one  of  Couette's  tubes,  the  apparent 
viscosity  (77)  is  0.01389.  From  the  values  of  17  and  ?7C  the  value 
of  A  may  be  calculated  as  above.  It  is  0.32  cm  and  the  diameter 


FIG.  4. — Capillary-tube  viscometer.     Couette. 

of  the  tube  is  0.090  cm  so  that  the  fictitious  length  to  be  added 
is  a  little  over  three  times  the  diameter  of  the  tube. 

In  the  experiments  used  by  Couette  to  calculate  the  value 
of  A  the  kinetic  energy  correction  is  very  large,  hence  a  consider- 
able error  may  have  been  introduced  by  taking  m  as  equal  to 
1.00  instead  of  the  more  probable  1.12.  Furthermore  the  range 
of  data  used  in  establishing  his  conclusion  is  rather  limited. 
Hence,  Knibbs  has  made  an  extended  study  of  the  same  subject. 
If  for  A  we  substitute  nR,  Eq.  (9)  may  be  written 


26 


FLUIDITY  AND  PLASTICITY 


but  since  from  Eq.  (9a)  we  have  that 
f          mpV2  \ 

J>  ~  w&> '  =  ° 


and  therefore 


If  values  of  ~oyj    are 


>5 


FIG.  5. — Finding  the  value  of 
n  for  the  "end  correction." 


This  is  the  equation  of  a  straight  line. 

plotted  as  ordinates  and  those  of  R/l  as  abscissas,  the  intercept 
on  the  axis  of  ordinates  will  give  the  corrected  viscosity,  i.e.,  the 
value  of  the  viscosity  when  I  =  •»  or  R  =  0;  and  the  tangent  of 
the  angle  made  by  the  line  with  the  axis  of  abscissas  when  divided 
by  the  viscosity  will  give  the  factor 
n  required.  Figure  5,  taken  from 
Knibbs'  work,  illustrates  the  method 
as  applied  to  the  tubes  used  by  Poi- 
seuille  B  to  Bv  and  F  to  FIV.  The 
values  of  n  are  found  to  be  —5.2  and 
+  11.2  respectively.  According  to 
Knibbs  "these  results  challenge  the 
propriety  of  Couette's  statement  that 
A  may  be  always  regarded  as  positive 
and  taken  as  nearly  three  times  the 
diameter  of  the  tube."  In  order  to  adequately  test  the  question 
Knibbs  took  the  whole  series  of  Poiseuille's  experiments  at  10° 
and  reduced  them  rigorously  on  the  basis  of  Eq.  (8)  taking  into 
account  the  peculiarities  of  the  bore  of  the  tubes  used  by  Poiseuille 
as  indicated  in  his  data.  Whenever  possible  the  value  of  pt  (cf. 
Eq.  (9))  was  obtained  by  extrapolation  since  then  the  correction 
term  vanishes;  in  the  other  cases  marked  with  a  star,  the  value 
of  m  was  taken  as  1.12.  The  results  are  arranged  according  to 
increasing  values  of  R/l,  since  if  n  has  a  positive  value  there 
should  be  a  progressive  increase  in  the  values  of  the  viscosity. 

Rejecting  the  last  four  values  as  uncertain,  the  general  mean  is 
0.013107  which  is  almost  identical  with  the  mean  for  each  group 
of  eight,  whereas  if  n  had  a  constant  value  there  should  be  a 
steady  progression.  On  the  other  hand  the  values  for  the  vis- 
cosity for  the  B  series  of  tubes  increase  while  those  for  the  F  series 
decrease  as  we  go  down  the  Table.  It  appears  therefore  that  no 
general  value  can  be  assigned  to  n  unless  it  be  zero. 


AMPLIFICATION  OF  THE  LAW  OF  PO I  SEVILLE 


27 


TABLE  IX. — THE  VISCOSITY  OF  WATER  AT  10°  CALCULATED  BY  KNIBBS  FROM 
POISETJILLE'S  EXPERIMENTS,  USING  EQ.  (8) 


Tube 

f  X  10S 

R*  X  1010 

i? 

D 

22 

0.242840 

0.013074* 

M  
C  
D1  
B  
C1  
E  
A  

B1  
F  
C"  
D11  
A1 

37 
42 
44 
56 
57 
64 
70 

75 
85 
86 
87 
93 

0.002367 
3.250400 
0.233770 
10.235000 
3.265900 
0.047160 
24.941000 
Mean  
10.276000 
11,207.000000 
3  .  298000 
0.227870 
25  059000 

0.013090* 
0.013028* 
0.013020* 
0.013202 
0.013071* 
0.013242* 
0.013145 
0.013109 
0.013134* 
0.013147 
0.013151* 
0.013078* 
0  013109* 

En 

115 

10  303000 

0  013070* 

A" 

139 

25  183000 

0  013119* 

F1  

163 

11,187.000000 
Mean 

0.013065 
0  013109 

E1  
C111  

174 
175 

0.048400 
3  339400 

0.013588* 
0  013092* 

Dm 

219 

0  224400 

0  013045* 

Bin 

240 

10  331000 

0  013002* 

A111 

277 

25  231000 

0  012946 

Fn 

326 

11  233  000000 

0  013249 

QIV 

421 

3  339400 

0  012498* 

AIV  

450 

25  231000 

0  013343 

Mean 

0  013095 

M1  
Biv 

558 
630 

0.002367 
10  357000 

0.013181* 
0  012742 

F"1 

646 

11  290  000000 

0  013967 

DIV  
E1  
Cv  
Av  
AVI  

jpiv 

649 

706 
709 
742 
1,046 

1  254 

0.223310 
0.048400 
3.339400 
25.231000 
25.231000 
Mean  
11  316  000000 

0.012652* 
0.013222* 
0.012015 
0.013515 
0.013607 
0.013113 
0  014891 

Bv 

1  455 

10  368000 

0  012193 

Fv 

3  034 

11  316  000000 

0  014851 

Avn... 

7,088 

25.231000 

0.016980 

28 


FLUIDITY  AND  PLASTICITY 


Bingham  and  White  (1912)  have  confirmed  the  conclusion  of 
Knfbbs  by  a  study  of  interrupted  flow.  A  capillary  I  =  9.38  cm 
R  =  0.01378  cm  was  used  to  determine  the  time  of  flow  of  a 
given  volume  of  water  at  25°  under  a  determined  pressure.  The 
capillary  was  then  broken  squarely  in  two  and  the  parts  separated 
by  glass  tubing,  the  whole  being  afterward  covered  with  stout 
rubber  tubing.  The  time  of  flow  was  again  determined  under 
the  same  conditions  as  before  except  that  the  corrections  for 
kinetic  energy  and  for  the  effects  of  the  ends  of  the  tubes  were 
doubled  by  the  interruption  in  the  flow.  The  breaking  of  the 
capillary  was  then  repeated  until  the  capillary  was  in  six  parts, 
the  corrections  necessary  being  proportional  to  the  number  of 
capillaries.  For  this  case  Eq.  (10)  becomes 
•jrgR*pt  mpVb 


+  &A) 

nt 

-  C' 


6A) 


mb 


(12) 


"  I  +  6A  /  +  6A 

where  C  and  C'  are  constants  under  the  conditions  of  experiment, 
and  b  is  the  number  of  capillaries,  and  A  as  before  is  the  fictitious 
length  to  be  added  to  each  capillary.  Substituting  in  Eq.  (12) 
the  values  of  the  time  of  efflux  and  the  pressure  when  the  capillary 
is  unbroken  ti  and  p\  and  when  broken  t2  and  p2  respectively, 
we  obtain  the  relation 

I  +  6A       Cp2tz  - 


hence, 


I  +  A 


A  = 


ti  -  C'm/ti 
K  -  1 


K 


b  -  K 


I. 


TABLE  X. — EXPERIMENTS  TO  DETERMINE  THE  "FICTITIOUS  LENGTH" 
OF  A  CAPILLARY  UNDER  CONDITIONS  OF  INTERRUPTED  FLOW 


Number  of 
capillaries  b 

Time 

Pressure  in 
grams 
per  cm2 

Cvt    i.iac'6 

K 

A 

cPt        t 

I 

179.7 

87.46 

0.0836 

2 

180.2 

87.77 

0.0837 

1.001        0.009 

3 

182.4 

87.32 

0.0835 

0.999 

0.006 

4 

183.1 

87.75             0.0836 

1.000 

0.000 

6 

185.0 

88  .  25              0  .  0838 

1.002 

0.003 

AMPLIFICATION  OF  THE  LAW  OF  POI SEVILLE  29 

Since  the  values  of  K  are  unity  within  the  experimental  error  the 
addition  to  the  length  is  zero.  In  no  single  instance  does  the 
value  of  A  amount  to  even  one-half  the  diameter  of  the  tube.  If 
however  the  value  of  m  had  been  taken  as  unity,  A  would  have 
appeared  to  have  positive  value. 

Had  A  been  found  to  have  a  definite  value,  it  would  have 
been  necessary  to  consider  the  legitimacy  of  making  the  correc- 
tion by  means  of  an  addition  to  the  length  of  the  capillary  instead 
of  by  means  of  a  correction  in  the  pressure  as  suggested  in  Eq.  (2), 
but  since  no  definite  value  can  be  assigned  to  this  correction, 
there  is  no  need  for  raising  the  question. 

The  shape  of  the  ends  of  the  tube  are  of  considerable  impor- 
tance in  determining  the  development  of  turbulent  flow,  under  cer- 
tain conditions.  Tubes  with  trumpet-shaped  entrances  appear  to 
promote  linear  flow  (cf.  Reynolds  (1883)  and  Couette(  1890)  p.  486). 

Slipping. — Coulomb  (1801)  made  experiments  with  an  oscillat- 
ing disk  of  white  metal  immersed  in  water,  and  he  noted  that 
coating  the  disk  with  tallow  or  sprinkling  it  over  with  sandstone 
had  no  effect  upon  the  vibrations.  This  seemed  to  prove  that  the 
fluid  in  contact  with  the  disk  moved  with  it,  and  that  the  property 
being  measured  was  characteristic  of  the  fluid  and  not  of  the 
nature  of  the  surface.  These  observations  were  confirmed  by 
O.  Meyer  in  1861. 

After  the  Law  of  Poiseuille  had  been  experimentally  and 
theoretically  established,  it  was  still  unsatisfactory  that  the 
results  of  measurements  of  viscosity  by  the  efflux  method  did 
not  agree  with  those  by  other  methods.  It  was  natural  to 
suppose  that  the  discrepancy  might  be  explained  by  the  external 
friction  between  the  fluid  and  the  solid  boundary  which  had  been 
assumed  by  Navier  (1823),  cf.  also  Margules  (1881)  and  Hada- 
mard  (1903).  Helmholtz  in  his  derivation  of  the  Law  of  Poi- 
seuille had  taken  into  account  the  effect  of  slipping  and  obtained 
the  formula,  which  in  our  notation  is 


where  X  depends  upon  the  nature  of  the  fluid  as  well  as  upon  that 
of  the  bounding  surface.  In  treatises  on  hydrodynamics  this  is 
usually  written 

F-g 


30  FLUIDITY  AND  PLASTICITY 

/3  being  the  coefficient  of  sliding  friction  which  is  the  reciprocal 
of  the  coefficient  of  slipping. 

From  the  experiments  of  Piotrowski  upon  the  oscillations 
of  a  hollow,  polished  metal  sphere,  suspended  bifilarly  and  filled 
with  the  viscous  liquid,  Helmholtz  deduced  a  value  for  X  of 
0.23534  for  water,  but  it  is  worth  noting  that  he  deduced  a 
value  of  the  viscosity  which  was  about  40  per  cent  greater  than 
that  obtained  by  the  efflux  method.  From  some  efflux  experi- 
ments of  Girard  (1815)  using  copper  tubes,  Helmholtz  deduced 
the  value  X  =  0.03984.  More  recently  Brodman  (1892)  has 
experimented  with  concentric  metal  spheres  and  coaxial  cylin- 
ders, the  space  between  being  filled  with  the  viscous  substance. 
He  thought  that  he  found  evidence  of  slipping. 

Slipping  can  be  best  understood  in  cases  where  a  liquid  does 
not  wet  the  surface,  as  is  true  of  mercury  moving  over  a  glass 
surface.  If  we  consider  a  horizontal  glass  surface  A,  Fig.  6,  as 
being  moved  tangentially  toward  the  right  over  a  surface  E, 


FIG.  6. 

between  which  there  is  a  thin  layer  of  mercury  C,  then  we  can 
imagine  that  the  mercury  is  separated  from  the  glass  on  either 
side  by  thin  films  B  and  D  of  some  other  medium,  usually  air. 
Points  in  a  surface  at  right  angles  to  the  above  indicated  by 
abed  may  at  a  later  time  occupy  the  relative  positions  a'b'c'd  or  if 
the  films  B  and  D  are  more  viscous  than  the  mercury  the  section 
may  be  better  represented  by  a"b"c"d.  But  from  Eq.  (1) 

dv  a  (f>ds 

so  that  in  any  case,  the  respective  contributions  to  the  flow  by  the 
inner  mercury  layer  or  by  the  superficial  films  will  depend  upon 
their  relative  fluidities  and  their  relative  thicknesses.  Whether 
the  liquid  wets  the  surface  or  not,  anything  which  affects  the 


AMPLIFICATION  OF  THE  LAW  OF  POISEUILLE  31 

fluidity  of  the  surface  film,  whether  it  be  surface  tension,  absolute 
pressure,  positive  or  negative  polarization,  static  electricity,  or 
magnetism  may  therefore  affect  the  amount  of  flow.  And  these 
effects  when  detected  experimentally  would  undoubtedly  be 
attributed  to  slipping  or  to  the  overcoming  of  external  friction. 
So  while  we  might  expect  the  effect  of  slipping  to  be  more  pro- 
nounced in  cases  where  the  liquid  does  not  wet  the  surface,  it  is 
quite  possible  that  even  when  the  liquid  does  wet  the  surface, 
the  fluidity  of  the  liquid  near  the  surface  is  not  identical  with 
that  within  the  body  of  the  liquid. 

On  the  other  hand,  it  is  important  to  remember  that  the 
thickness  of  the  layer  of  liquid  affected  by  the  forces  of  adhesion, 
with  which  we  are  here  chiefly  concerned,  is  only  molecular. 
Even  with  mercury  in  a  glass  tube,  the  thickness  of  the  layer  of 
air  seems  to  be  of  molecular  dimensions.  One  may  get  an  idea  of 
the  upper  limit  to  this  thickness  by  the  following  experiment. 
A  thread  of  mercury  was  placed  in  a  narrow  capillary  so  that 
the  air  surface  would  be  relatively  large.  Taking  care  that  no 
air-bubbles  were  present,  the  length  of  the  thread  was  measured 
with  a  dividing  engine,  in  a  determined  part  of  the  tube.  The 
tube  was  exhausted  from  both  ends  simultaneously  and  the 
thread  moved  back  and  forth  in  order  to  sweep  out  the  supposed 
layer  of  air.  When  the  mercury  was  finally  brought  back 
to  its  former  position  no  decrease  in  length  could  be  detected. 
In  order  to  have  slipping  under  ordinary  conditions  of  measurement 
it  would  appear  that  the  surface  film  must  be  of  very  much  more 
than  molecular  thickness  or  else  it  must  have  practically  infinite 
fluidity.  In  view  of  the  strong  adhesion1  between  all  liquids  and 
solids  it  seems  improbable  that  the  particular  layer  of  liquid  in 
contact  with  the  solid  should  show  an  amount  of  flow  which  is 
comparable  in  amount  with  that  of  all  of  the  other  practically 
infinite  layers  of  liquid. 

Nevertheless  if  the  value  deduced  by  Helmholtz  for  water 
on  a  metal  surface  be  correct,  X  =  0.23534,  the  effect  of  slipping 
ought  to  be  readily  observed.  According  to  Whetham  (1890),  if 
we  take  R  =  0.051,  Eq.  (12)  becomes 

V  =  ^  117.67  X  10~6 
lCf.  Duclaux,  1872. 


32  FLUIDITY  AND  PLASTICITY 

whereas  if  there  were  no  slip  and  therefore  X  =  0  we  would  have 

F  =  ^7  6.25  X  10-6 
8rjl 

Thus  it  would  appear  that  the  rate  of  flow  through  a  polished 
metal  tube  should  be  nearly  20  times  more  rapid  than  through  a 
tube  in  which  there  is  no  slip.  Since  Poiseuille's  experiments 
prove  that  the  viscosity  is  constant  for  tubes  of  very  different 
radius  when  calculated  without  regard  to  slipping,  there  can  be  no 
slipping  when  water  flows  through  glass  tubes.  This  conclusion 
is  admitted  by  Helmholtz. 

Jacobson  (1860)  criticised  Helmholtz's  use  of  Girard's  experi- 
ments in  that  he  failed  to  apply  any  correction  to  the  pressure. 
Jacobson  himself  experimented  with  copper  tubes  as  well  as  glass 
tubes  but  found  no  evidence  of  slipping. 

Warburg  (1870)  investigated  the  flow  of  mercury  in  glass 
tubes.  He  found  that  Poiseuille's  law  of  pressures  and  his  law  of 
diameters  were  verified,  which  proved  that  slipping  did  not  occur. 
Be"nard  as  reported  by  Brillouin  (1907)  page  152,  has  repeated 
the  work  of  Warburg  using  greater  care,  and  he  finds  that  X 
cannot  have  a  value  greater  than  0.00001. 

Whetham  (1890)  caused  water  to  flow  through  a  glass  tube 
before  and  after  being  silvered,  proper  corrections  being  made  for 
changes  in  temperature  and  in  the  radius  of  the  tube,  due  to 
the  silver  layer.  Different  thicknesses  of  silver  as  well  as 
different  pressures  were  used,  but  the  difference  in  the  times  of 
flow  between  the  silvered  and  unsilvered  tubes  were  all  within 
the  limit  of  experimental  error.  Copper  tubes  were  also  used  and 
the  results  in  all  cases  were  in  agreement  with  Poiseuille's 
observations.  Cleaning  the  tubes  with  acids  and  alkalies,  polish- 
ing with  emery  powder,  coating  with  a  film  of  oil  and  amalgamat- 
ing with  mercury  were  all  without  effect  in  producing  a  deviation 
which  could  be  detected.  Whetham  repeated  an  experiment  of 
Piotrowski  with  an  oscillating  glass  flask,  plain  and  silvered. 
Care  was  taken  to  make  correction  for  temperature  and  to 
prevent  changes  in  the  bifilar  suspension  which  seems  to  have 
been  neglected  by  Piotrowski.  Whetham  found  the  ratio  of  the 
friction  of  water  on  glass  to  the  friction  of  water  on  silver  to  be 
1.0022,  which  may  be  taken  as  unity  within  the  limits  of  experi- 
mental error.  Couette  (1888-1890)  attacked  the  problem 


AMPLIFICATION  OF  THE  LAW  OF  PO I  SEVILLE 


33 


independently  but  along  much  the  same  lines.  He  tried  the 
effect  of  a  layer  of  grease  and  of  silver  on  the  inside  of  a  tube. 
He  found  invariably  the  same  efflux  time  or  even  a  little  greater 
which  was  due  to  the  diminution  in  the  radius  of  the  tube.  But 
even  this  latter  effect  did  not  occur  when  the  thickness  of  the 
silver  layer  was  a  negligible  fraction  of  the  radius  of  the  tube. 
He  then  used  tubes  of  white  metal,  copper,  and  paraffin  using 
rates  of  efflux  close  to  the  critical  values,  and  obtained  the 
following  results: 

TABLE  XI. — COUETTE'S  EXPERIMENTS  ON  SLIPPING 


Substance  of  tube 

Temperature 

,        1  T]  Calculated  from 
Tj  Observed                 „  .       .,, 
roiseuille 

Copper  
Copper  
White  metal  

15.5 
17.3 
18.2 

0.01175 
0.01073 
0.01037 

0.01130 
0.01079 
0.01055 

White  metal  

18.9 

0.01064 

0.01037 

White  metal  

18.3 

0.01092 

0.01052 

Paraffin  

12.6 

0.01241 

0.01219 

Paraffin  

12.9 

0.01278 

0.01209 

Paraffin  

12.3 

0.01276 

0.01228 

Couette  goes  further  and  gives  reasons  for  the  conclusion  that 

slipping  does  not  occur  even  after  the  flow  becomes  turbulent. 

More  recently  Ladenburg  (1908)  has  carefully  repeated  the 

experiments  of  Piotrowski  under  as  nearly  as  possible  the  same 

TABLE   XII. — LADENBURG'S   EXPERIMENTS   WITH   AN   OSCILLATING   GLASS 
FLASK,  SHOWING  ABSENCE  OF  SLIPPING,  AT  19 . 0° 


Flask 

Logarithm  dec- 
rement 

Period    of 
vibration 

Remark 

A  

0.019570+2 

11,973±2 

Unsilvered 

A  

0.019642+3 

12,049+2 

Silvered 

A  

0.019620  ±3 

11,990  +  1 

Unsilvered          j 

B  

0.025026+25 

11,716+4 

Unsilvered 

B  

0.025011  +  15 

11,688+2 

Silvered 

C  

0.025162  ±2 

11,870±2 

Silvered 

34 


FLUIDITY  AND  PLASTICITY 


experimental  conditions.  He  used  plain  and  silvered  oscillating 
glass  vessels  and  a  hollow  metal  sphere.  Table  XII  proves 
conclusively  that  slipping  was  absent  in  the  former  case. 

Using  the  hollow  metal  sphere  filled  with  water,  Ladenburg 
obtained  values  of  the  viscosity  which  agree  with  the  values  found 
by  other  methods,  and  shown  in  Table  XIII. 

TABLE  XIII. — A  COMPARISON  OF  THE  VISCOSITY  OF  WATER  AS  OBTAINED 
BY  DIFFERENT  METHODS  (LADENBURG) 


Method 

17  at  17.5° 

,  at  19.2° 

Observer 

Efflux  glass 

0.01076 

0.01031 

Poiseuille  (1846) 

Efflux  glass  
Efflux  glass  
Efflux  glass  
Oscillating  solid  sphere  
Oscillating  hollow  cylinder.  . 
Oscillating  hollow  sphere  .  .  . 

0.01065 
0.01075 
0.01067 
0.01099 
0.01082 
0.01065 

0.01027 
0.01030 
0.01025 
0.01054 
0.01037 
0.01032 

Sprung  (1876) 
Slotte  (1883) 
Thorpe  and  Rodger  (1894) 
W.  Konig  (1887) 
Mutzel  (1891) 
Ladenburg  (1908) 

Ladenburg  indicates  how  Helmholtz  erroneously  obtained 
his  large  coefficient  of  slipping  by  overlooking  a  point  in  the 
theory,  and  recalculating  Piotrowski's  data  he  finds  that  instead 
of  the  viscosity  being  40  per  cent  greater  than  the  generally 
accepted  value,  this  difference  becomes  only  3  per  cent  and  the 
slipping  becomes  negligible. 

It  was  stated  above  that  the  verification  of  the  Law  of 
Diameters  of  Poiseuille  is  a  proof  that  slipping  does  not  occur 
between  glass  and  water.  Knibbs  (1895)  has  collected  an 
extensive  table  of  observations  of  the  viscosity  of  water  at  10° 
for  tubes  of  various  materials  having  radii  varying  from  0.0140  to 
0.6350  cm  or  nearly  a  thousand-fold,  but  there  is  no  evidence  of 
progressive  deviation  as  the  radius  increases. 

In  experimenting  on  the  possible  effect  of  an  electrical  or 
magnetic  field  upon  viscosity,  W.  Konig  (1885)  obtained  a 
negative  result.  Duff  (1896)  seemed  to  detect  an  increase  in  the 
viscosity  of  castor  oil  of  0.5  per  cent  using  the  falling  drop 
method  and  a  potential  gradient  of  27,000  volts  per  centimeter, 
but  for  the  most  part  the  results  were  negative.  Quincke  (1897) 


AMPLIFICATION  OF  THE  LAW  OF  POI SEVILLE  35 

found  a  definite  effect  on  the  viscosity  in  an  electrical  field,  which 
Schaufelberger  (1898)  attempted  to  explain  on  the  basis  of 
hysteresis.  However,  Pacher  and  Finazzi  (1900)  obtained  results 
which  were  contrary  to  those  of  Duff  and  Quincke  finding  that 
insulating  liquids  under  the  action  of  an  electrical  field  do  not 
undergo  any  sensible  change  in  viscosity.  Ercolini  (1903)  made 
experiments  along  the  same  line  and  concluded  that  the  effect 
was  less  than  his  experimental  error.  He  used  petroleum, 
benzene,  turpentine,  olive  oil,  and  vaseline.  Carpini  (1903) 
measured  the  viscosity  of  magnetic  liquids  in  a  magnetic  field  but 
found  no  certain  effect.  Koch  (1911)  tried  the  effect  of  oxygen 
or  hydrogen  polarization  at  the  boundary  using  a  platinum  tube 
and  an  oscillating  copper  disk.  No  change  in  the  viscosity  was 
observed  and  Koch  regards  this  as  strong  evidence  against 
slipping.  Ronceray  (1911)  has  studied  the  effect  of  surface 
tension. 

These  results  seem  to  make  it  quite  certain  that,  whether 
the  liquid  wets  the  solid  or  not,  there  is  no  measurable  difference 
between  the  velocity  of  the  solid  and  of  the  liquid  immediately  in 
contact  with  it,  at  least  so  long  as  the  flow  is  linear. 

The  Transition  from  Linear  to  Turbulent  Flow. — It  is  well 
known  that  the  formulas  which  have  been  discussed  do  not 
apply  to  the  ordinary  flow  of  liquids  in  pipes.  Under  ordinary 
conditions  we  know  that  the  flow  is  undulatory,  instead  of  being 
linear  as  is  assumed  in  the  simple  laws  of  motion.  It  is  important 
that  we  know  under  what  conditions  these  sinuous  motions 
appear  so  that  they  may  be  properly  taken  into  account  or 
guarded  against.  An  extended  study  of  the  flow  of  water  in 
pipes  having  a  diameter  varying  from  0.14  to  50  cm  was  made  by 
Darcy  (1858).  He  found  the  hydraulic  resistance  proportional 
to  In  where  n  had  a  value  nearly  equal  to  2  (1.92).  He  saw  more 
clearly  than  any  of  his  predecessors  that  hydraulic  flow  is  very 
different  in  character  from  the  viscous  flow  studied  by  Poiseuille, 
since  the  viscous  resistance  is  proportional  to  the  first  power  of 
the  mean  velocity  (/).  Darcy  paid  little  attention  to  the  tem- 
perature at  which  his  experiments  were  carried  out,  probably  as 
Reynolds  remarks,  because  "the  resistance  after  eddies  have 
been  established  is  nearly,  if  not  quite,  independent  of  the 
viscosity."  Since  Darcy's  work  was  approved  by  the  Academy 


36 


FLUIDITY  AND  PLASTICITY 


in  1845,  he  is  probably  the  first  to  distinguish  clearly  between 
the  two  regimes. 

Hagen  (1854)  investigated  the  effect  of  changes  in  temperature 
upon  the  rate  of  efflux  in  tubes  of  moderate  diameters.     Figure  7 


8.?! 


•NARROW  TUBE 


MEAN  TUBE 

WIDE  TUBE 

III! 


0.71 
0.98 


~0         5        10       15       ?0      15      10      35       40  ,  45       50      55       60      65      10 
Temperature  Degrees  Reaumur 

FIG.  7. — Transition  from  linear  to  turbulent  flow.     The  effect  of  temperature. 


exhibits  the  results  of  his  experiments.  The  abscissas  are 
degrees,  Reaumur,  the  ordinates  the  volumes  in  cubic  inches 
("Rheinland  Zollen  ")  transpiring  in  a  unit  of  time.  The  pressure 
to  which  each  curve  corresponds  is  given  at  the  right  of  the  figure, 
being  expressed  in  inches  of  water.  Hagen  used  three  tubes  of 
varying  width  as  follows: 


AMPLIFICATION  OF  THE  LAW  OF  POI SEVILLE  37 


Tube 

Radius,  inches 

Length,  inches 

Narrow 

0  053844 

18  092 

Mean. 

0  077394 

41  650 

Wide  

11.391400 

39.858 

FIG.  8. — Apparatus  of  Reynolds  for  studying  the  critical  regime. 

Inspection  of  the  figure  shows  that  with  the  lowest  pressure  and 
the  smaller  tubes  the  efflux  is  a  linear  function  of  the  temperature 
except  at  the  highest  temperatures.  With  the  wide  tube,  however, 
there  is  a  maximum  of  efflux  at  about  37°  even  at  the  smallest 
pressure.  As  the  pressure  is  increased  the  maximum  appears  at  a 


38  FLUIDITY  AND  PLASTICITY 

lower  and  lower  temperature  and  the  maximum  appears  even  in 
the  smallest  tube  used.  There  is  a  minimum  of  efflux  after 
passing  the  maximum  but  then  the  efflux  becomes  again  a  linear 
function  of  the  temperature.  Brillouin  (1907)  page  208,  has 
confirmed  the  experimental  results  of  Hagen. 

A  clear  picture  of  the  phenomena  connected  with  the  passage 
from  one  regime  to  the  other  has  been  given  by  Reynolds  (1883). 
One  form  of  apparatus  used  by  him  is  depicted  in  Fig.  8.  It 


FIG.  9.— Linear  flow. 

consists  of  a  glass  tube  BC,  with  a  trumpet-shaped  mouthpiece 
AB  of  wood,  which  was  carefully  shaped  so  that  the  surfaces 
would  be  continuous  from  the  wood  to  the  glass.  Connected 
with  the  other  end  is  a  metal  tube  CD  with  a  valve  at  E  having 
an  opening  of  nearly  1  sq.  in.  The  cock  was  controlled  by  a  long 
lever  so  that  the  observer  could  stand  at  the  level  of  the  bath, 
which  surrounded  the  tube  BC.  The  wash-bottle  W  contained 
a  colored  liquid  which  was  led  to  the  inside  of  the  trumpet- 
shaped  opening.  The  gage  G  was  used  for  determining  the  level 


Fia.  10. — The  beginning  of  turbulent  flow. 

of  water  in  the  tank.  When  the  valve  E  was  gradually  opened 
and  the  color  was  at  the  same  time  allowed  to  flow  out  slowly,  the 
color  was  drawn  out  into  a  narrow  band  which  was  beautifully 
steady  having  the  appearance  shown  in  Fig.  9.  Any  consider- 
able disturbance  of  the  water  in  the  tank  would  make  itself 
evident  by  a  wavering  of  the  color  band  in  the  tube;  sometimes  it 
would  be  driven  against  the  glass  tube  and  would  spread  out, 
but  without  any  indication  of  eddies. 

As  the  velocity  increased  however,  suddenly  at  a  point  30  or 
more  times  the  diameter  of  the  tube  from  the  entrance,  the  color 


AMPLIFICATION  OF  THE  LAW  OF  POISEUILLE  39 

band  appeared  to  expand  and  to  fill  the  remainder  of  the  tube 
with  a  colored  cloud.  When  looked  at  by  means  of  an  electric 
spark  in  a  darkened  room,  the  colored  cloud  resolved  itself  into 
distinct  eddies  having  the  appearance  shown  in  Fig.  10.  By 
lowering  the  velocity  ever  so  slightly,  the  undulatory  movement 
would  disappear,  only  to  reappear  as  soon  as  the  velocity  was 
increased.  If  the  water  in  the  tank  was  not  steady  the  eddies 
appeared  at  a  lower  velocity  and  an  obstruction  in  the  tube 
caused  the  eddies  to  be  produced  at  the  obstruction  at  a  consider- 
ably lower  velocity  than  before.  "Another  phenomenon  which 
was  very  marked  in  the  smaller  tubes  was  the  intermittent  char- 
acter of  the  disturbance.  The  disturbance  would  suddenly  come  on 
through  a  certain  length  of  the  tube,  pass  away,  and  then  come 
again,  giving  the  appearance  of  flashes,  and  these  flashes  would 
often  commence  successively  at  one  point  in  the  pipe."  The  ap- 
pearance when  the  flashes  succeeded  each  other  rapidly  is  shown 


w 


FIG.   11. — Flashing. 

in  Fig.  11.  "This  condition  of  flashing  was  quite  as  marked 
when  the  water  in  the  tank  was  very  steady,  as  when  somewhat 
disturbed.  Under  no  circumstances  would  the  disturbance  occur 
nearer  the  funnel  than  about  30  diameters  in  any  of  the  pipes, 
and  the  flashes  generally,  but  not  always  commenced  at  about 
this  point.  In  the  smaller  tubes  generally,  and  with  the  larger 
tube  in  the  case  of 'ice-cold  water  at  4°,  the  first  evidence  of 
instability  was  an  occasional  flash  beginning  at  the  usual  place 
and  passing  out  as  a  disturbed  patch  2  or  3  in.  long.  As  the 
velocity  further  increased  these  flashes  became  more  frequent 
until  the  disturbance  became  general." 

Reynolds  further  noted  that  the  free  surface  of  a  liquid  indi- 
cates the  nature  of  the  motion  beneath.  In  linear  flow,  the  sur- 
face is  like  that  of  plate  glass,  in  which  objects  are  reflected  without 
distortion,  while  in  sinuous  flow,  the  surface  is  like  that  of  sheet 
glass.  A  colored  liquid  flowing  out  into  a  vessel  of  water  has  the 
appearance  of  a  stationary  glass  rod  in  the  first  regime,  but  as  the 


40 


FLUIDITY  AND  PLASTICITY 


velocity  is  increased  the  surface  takes  on  a  sheet  glass  appearance 
due  to  the  sinuous  motions,  and  finally  the  stream  breaks  into 
eddies  and  is  lost  to  view  (cf.  Collected  Papers  2,  158). 

Reynolds  reasoned  from  the  equations  of  motion  that  the 
birth  of  eddies  should  depend  upon  a  definite  value  of 


where  R  is  a  single  linear  parameter,  as  the  radius  of  the  tube, 
and  /  is  a  single  velocity  parameter,  as  the  mean  velocity  of  flow 
along  the  tube.  Reynolds  found  the  value  of  the  constant  to  be 
approximately  1,000,  hence,  the  maximum  mean  velocity  in 
centimeters  per  second  for  which  we  may  expect  linear  flow,  may 
be  taken  to  be 

/  =  im  (14) 

In  Table  XIV  we  have  calculated  the  value  of  the  product 
pRI<p  from  some  of  Reynolds'  observations  near  the  critical 
velocity. 

TABLE  XIV. — CALCULATION  OF  THE  CRITICAL  VELOCITY  CONSTANT 


Flow 

Temper- 
ature 

P 

R  centi- 
meters 

V 

I  centi- 
meters per 
second 

pRI9 

Steady  

9 

1.00 

0.3075 

74.4 

44.26 

,012 

Unsteady  .  .  . 

8 

1.00 

0.3075 

72.4 

48.65 

,113 

Steady  

5 

1.00 

0.3075 

66.2 

51.06 

,039 

Unsteady... 

5 

1.00 

0.3075 

66.2 

54.33 

,106 

Steady  

8 

1.00 

0.6350 

72.4 

22.60 

,039 

Unsteady.  .  . 

8 

1.00 

0.6350 

72.4 

22.60 

,039 

Reynolds  tried  plotting  the  mean  velocity  against  the  fall  in 
pressure  per  unit  length  of  the  tube  as  shown  in  Fig.  12.  It  is 
to  be  observed  that  the  resistance  increases  as  a  linear  function 
of  the  velocity  according  to  Poiseuille's  law  up  to  a  certain  definite 
point,  and  that  from  that  point  on  the  resistance  varies  as  some 
higher  power  of  the  velocity.  This  power  as  we  shall  see  is 
constant  and  is,  according  to  Reynolds  equal  to  1.723  over  a 
wide  range  of  pressures.  Formulated  this  relation  becomes 

P  =  KI»  (15) 


AMPLIFICATION  OF  THE  LAW  OF  PO I  SEVILLE 


41 


where  P  is  the  pressure  gradient  and  n  is  the  constant.     In  the 
first  re'gime  n  —  1  and  we  have 

P  =  KI 
It  may  be  remarked  here  that  hydraulicians  have  usually 


elocity  in  Meters  per  Seconc 
§P 

01 

^ 

^ 

^ 

s 

^ 

x 

7.4 

.  " 

•^^ 

/ 

,' 

_^- 

•  — 

/ 

X 

/ 

/ 

/ 

I 

/ 

/ 

y 

0.0 


0.15 


0.05  0.10 

Slope  of  Pressure  'in  U/crter 

FIQ.  12. — Efflux-shear  curves  passing  through  the  critical  regime. 


EN  0.5 


-3 


-z  -\ 

Log  Slopes  of  Pressures 


FIG.  13. — A  better  method  for  bringing  out  the  characteristics  of  the  viscous, 
hydraulic,  and  critical  regimes. 


employed  the  expressions 


P  =  KP 


or 


P  =  AI  +  BI* 

t* 

Putting  Eq.  (15)  in  the  form 

log  P  =  n  log  /  +  log  K 

we  observe  that  the  relations  may  be  presented  more  forcefully  by 
plotting  the  logarithms  of  the  pressure  gradients  as  abscissas 


42  FLUIDITY  AND  PLASTICITY 

and  the  logarithms  of  the  mean  velocities  as  ordinates.  For 
tubes  4  and  5,  Reynolds  obtained  the  curves  given  in  Fig.  13. 
Linear  flow  exists  along  the  line  ABC,  or  A'B'C',  the  hydraulic 
regime  exists  along  the  line  BDEF  or  B'D'E'F'.  It  is  evident  now 
that  n  is  constant  for  each  part  of  the  curves  for  the  two  tubes 
and  that  it  is  the  same  for  both,  i.e.  the  curves  can  be  exactly 
superimposed  by  merely  a  rectangular  shift.  The  line  ABC  is 
inclined  at  an  angle  of  45°  so  that  n  =  1,  and  the  line  BDEF  is 
inclined  at  an  angle  30°-8'  so  that  n  =  1.723.  Except  for 
the  unstable  region  BCD,  the  formula  P  =  KIn  will  represent  the 
viscosity  in  both  regimes,  it  being  necessary  to  merely  change 
the  value  of  n  in  passing  from  one  regime  to  the  other. 

In  passing  from  B  to  C  it  is  evident  that  the  linear  flow  becomes 
increasingly  unstable,  and  thus  is  explained  why  the  eddies 
appear  suddenly  and  full-fledged,  when  the  disturbance  is 
sufficiently  great.  The  more  undisturbed  the  liquid  is,  the 
farther  is  it  possible  to  go  from  B. 

The  points  along  the  curve  CD  (or  C'D')  correspond  to  the 
mixed  regime  where  the  flashing  occurs,  the  turbulent  movement 
alternating  with  the  linear.  Light  has  been  thrown  upon  the 
cause  of  the  flashing  by  Couette  (1890)  and  Brillouin  (1907), 
using  a  horizontal  tube  opening  directly  into  the  air.  At  high 
pressures  the  surface  of  the  jet  had  a  sheet-glass  appearance 
indicating  that  the  flow  was  hydraulic  but  the  amplitude  of  the 
jet  was  constant.  As  the  pressure  was  lowered  the  velocity  fell 
to  a  point  where  linear  flow  began.  But  as  the  resistance  was 
much  less  in  linear  flow,  the  velocity  increased  as  shown  by  the 
increased  amplitude  of  the  jet,  and  the  condition  of  hydraulic 
flow  was  reestablished.  The  rapidity  of  these  fluctuations 
gradually  increased  as  the  pressure  was  further  reduced,  passed 
through  a  maximum  and  gradually  declined  as  the  linear  flow 
came  to  predominate.  Finally  the  jet  became  regular  with  a 
plate  glass  surface. 

It  has  been  suggested  that  there  are  really  three  regimes,  one 
for  velocities  with  which  only  linear  flow  can  exist,  a  second  for 
velocities  with  which  only  turbulent  flow  can  exist,  and  a  third 
where  linear  and  turbulent  flow  alternate.  The  linear  regime  is 
sharply  marked  off  from  the  hydraulic  regime  by  the  point  B 
where  the  lines  ABC  ancj  BDE  intersect.  While  the  period  of 


AMPLIFICATION  OF 
—  <=>  "i 


THE  LAW  OF  POI SEVILLE 

eJ 


43 


\ 


\ 


Log  velocities 


44 


FLUIDITY  AND  PLASTICITY 


Number 

Diameter 

Temper- 
ature 

Surface 

A 

0.0014 
0.0270 
0.0650 
0.615 
1.270 
1.400 
2.700 
4.100 
2.600 
8.260 
19.600 
28.500 
8.190 
13.700 
18.800 
50.000 
24  .  320 
24.470 
4.968 

10 
10 
10 
5 
5 

12 
21 

21 
15 
15 

Glass  1 
Glass  ^   Poiseuille 
Glass  j 

Lead  No.  4  \ 
T      j  XT          f  rteyno 
Lead  No.  5  J 

Lead 
Lead 
Lead 
Varnished 
Varnished 
Varnished 
Varnished 
Cast  iron,  new 
Cast  iron,  new 
Cast  iron,  new 
Cast  iron,  new 
Cast  iron,  incrusted 
Cast  iron,  cleaned 
Glass 

ds 
Darcy 

B 

c 

D 

E      . 

F  
G  
H 

I 

J 

K 

L  
M  
N  

o 

p 

Q  
R 

S  

the  oscillations  in  the  mixed  regime  is  entirely  characteristic,  it 
seems  hardly  probable  that  we  can  always  sharply  differentiate  the 
mixed  from  the  hydraulic  regime.  Indeed  Couette  (p.  486) 


Fio.  15. — Coaxial  cylinder  viscometer  of  Couette. 

found  that  with  a  tapering  tube  the  oscillations  do  not  appear  at 
all.     One  may  draw  the  conclusion  from  Reynolds'  observations 


AMPLIFICATION  OF  THE  LAW  OF  POI SEVILLE 


45 


that  the  formula  P  =  KIn  may  be  used  in  the  critical  regime. 

Reynolds  has  compared  the  data  of  Darcy  for  large  tubes, 

that  of  Poiseuille  for  small  tubes,  with  his  own,  plotting  the 

logarithmic  homologues  as  in  Fig.  13.     The  result  is  shown  in 


U 


9' 


3/10 


FIG.  16. — Detail  of  coaxial  viscometer. 

Fig.  14.  Each  line  represents  the  logarithmic  homologue  for 
some  particular  tube,  described  in  the  figure.  It  is  at  once 
apparent  that,  for  the  most  part,  experiments  have  been  made 
well  below  or  else  well  above  the  critical  values.  In  the  small 
tubes  of  Poiseuille  the  velocities  were  below  the  critical  values. 


46 


FLUIDITY  AND  PLASTICITY 


The  smallest  tube  with  which  he  experimented,  A,  gives  a 
curve,  only  part  of  which  is  shown  in  the  figure.     It  should  be 


Ve  1  o  c  i  1 1) 
FIG.  17. — The  transition  from  viscous  to  hydraulic  flow  with  coaxial  cylinders. 

added  that  Reynolds  corrected  Poiseuille's  data  for  the  loss  in 
kinetic  energy. 

For  pipes  ranging  in  diameter  from  0.0014  to  500  cm  and  for 
pressure  gradients  ranging  from  1  to  700,000,  there  is  not  a 
difference  of  more  than  10  per  cent  in  the  experimental  and 


AMPLIFICATION  OF  THE  LAW  OF  POISEUILLE  47 

calculated  velocities  and,  with  very  few  exceptions,  the  agree- 
ment is  within  2  or  3  per  cent,  and  it  does  not  appear  that  there 
is  any  systematic  deviation. 

Couette  (1890)  has  strongly  confirmed  the  work  of  Reynolds 
by  his  measurements  with  coaxial  cylinders.  The  external 
appearance  of  the  apparatus  used  is  shown  in  Fig.  15  where  V 
is  the  outer  cylinder  of  brass  which  can  be  rotated  at  a  constant 
velocity  by  means  of  an  electric  motor  around  its  axis  of  figure  T. 
The  inner  cylinder  is  supported  by  a  wire  attached  at  n. 
A  section  through  a  part  of  the  apparatus  in  Fig.  16,  shows  the 
inner  cylinder  s  while  g  and  g'  are  guard  rings  to  eliminate  the 
effect  of  the  ends  of  the  cylinder.  The  torque  may  be  measured 
by  the  forces  exerted  on  the  pulley  r  which  are  necessary  to 
hold  the  cylinder  in  its  zero  position.  Plotting  viscosities  as 
ordinates  and  the  mean  velocities  as  abscissas,  he  obtained 
Fig.  17.  Curve  I  represents  the  results  for  the  coaxial  cylinders, 
curve  II  represents  the  same  results  on  five  times  as  large  a 
scale  in  order  to  show  better  the  point  where  the  regime  changes. 
Curves  III  and  IV  are  for  two  different  capillary  tubes.  It  is 
clear  from  the  figure  that  the  viscosity  is  quite  constant  up  to 
the  point  where  the  regime  changes.  The  apparent  viscosity 
then  increases  very  rapidly,  and  finally  becomes  a  linear  func- 
tion of  the  velocity.  The  dotted  parts  of  the  curves  where  the 
viscosity  increases  most  rapidly,  represents  the  region  of  the 
mixed  regime,  and  the  measurements  were  very  difficult  to  ob- 
tain with  precision. 

He  proved  that  pRI<p  =  a  constant  by  a  series  of  experiments. 

(1)  The  mean  velocity  at  the  lower  limit  of  the  oscillations  is 
independent  of  the  length  of  the  tube.  He  used  a  glass  tube 
R  =  0.1778  and  obtained  the  efflux  per  minute  V',  thus: 

TABLE  XV. — LAW  OF  LENGTHS 


Length,  centimeters 

V  mean 

86.5 

388 

71.5 

367 

57.9 

365 

41.8 

376 

25.7 

394 

48 


FLUIDITY  AND  PLASTICITY 


(2)  The  mean  velocity  at  the  lower  limit  of  the  oscillations  is 
inversely  proportional  to  the  radius  of  the  tube. 

TABLE  XVI. — LAW  OF  RADII 


R 

Temperature 

V 

V 
R 

0.04998 

12.7 

103.6 

2,073 

0.09036 

13.6 

214.9 

2,378 

0.13070 

13.6 

344.0 

2,632 

0.17780 

13.6 

377.0 

2,121 

0.21080 

13.6 

542.0 

2,570 

0.27620 

13.6 

701.0 

2,538 

0.29690  (Copper) 

15.0 

648.0 

2,182 

0.45000 

15.0 

1,205.0 

2,678 

(3)  The  mean  velocity  at  the  lower  limit  of  the  oscillations  is 
inversely  proportional  to  the  fluidity.     For  both  mercury  and 
water  an  elevation  of  the  temperature  caused  a  lowering  of  the 
mean  velocity  at  the  lower  limit  of  the  oscillations.     The  in- 
crease in  the  temperature  causes  an  increase  in  the  fluidity  in 
both  cases. 

(4)  Experiments  with  air  and  water  confirm  the  law  that  the 
mean  velocity  at  the  lower  limit  of  the  oscillations  is  inversely 
proportional  to  the  density  of  the  medium.     The  number  of 
turns  of  the  outer  cylinder  per  minute  is  taken  as  proportional 
to  the  mean  velocity,  a  being  a  constant. 

TABLE  XVII.— LAW  OP  DENSITIES 


Substance 

v 

P 

al 

(KppI 

Water  
Air 

0.01096 
0  00018 

1.0000 
0  0012 

56 
800 

5,100 
5,300 

There  would  be  still  some  doubt  whether  the  critical  velocity 
is  inversely  proportional  to  the  fluidity,  but  this  doubt  is  re- 
moved by  the  work  of  Coker  and  Clement  (1903)  to  test  this 
very  point.  They  used  a  single  tube  I  =  6  ft.  R  =  0.38  in. 
measuring  the  flow  of  water  over  a  range  of  temperatures  from 


AMPLIFICATION  OF  THE  LAW  OF  POI SEVILLE  49 

4°  to  nearly  50°.  Plotting  the  logarithmic  homologues  they 
obtained  a  family  of  curves  exactly  similar  to  those  in  Fig.  14, 
so  that  it  is  unnecessary  to  reproduce  them.  The  points  of 
intersections  between  the  curves  for  linear  and  for  turbulent 
flow  lie  on  a  perfectly  straight  line  as  is  true  in  Fig.  14.  This 
proves  that  the  critical  velocity  is  directly  proportional  to  the 
viscosity.  Indeed  plotting  the  critical  velocities  read  from  their 
curves  against  the  temperatures,  one  obtains  a  curve  which  is 
almost  identical  with  that  obtained  by  calculation  from  the 
viscosities  according  to  the  assumed  law. 

Compressible  Fluids. — As  a  compressible  fluid  flows  through 
a  capillary  under  pressure,  expansion  takes  place  as  the  pressure 
is  relieved.  The  expansion  may  give  rise  to  several  effects  which 
must  be  taken  into  consideration.  (1)  The  velocity  increases  as 
the  fluid  passes  along  the  tube.  (2)  There  must  be  a  component 
of  the  flow  which  is  toward  the  axis  of  the  tube.  (3)  The  expan- 
sion may  cause  a  change  of  temperature.  This  may  affect  the 
flow  in  two  ways  (a)  by  changing  the  volume  and  consequently 
the  velocity  and  (6)  by  changing  the  viscosity  of  the  medium  and 
consequently  the  resistance  to  the  flow.  (4)  As  the  density 
changes,  the  viscosity  may  also  change,  unless  the  viscosity  is 
independent  of  the  density.  (5)  We  must  also  consider  whether 
the  kinetic  energy  correction  is  changed  when  the  velocity 
increases  as  the  fluid  passes  along  the  tube. 

For  incompressible  fluids,  we  have  seen  that  the  viscosity 
measurement  may  be  made  without  reference  to  the  absolute 
pressure.  But  with  compressible  fluids  this  is  not  the  case, 
because  the  rate  of  expansion  depends  upon  the  absolute  pres- 
sures, in  the  two  reservoirs  at  the  level  of  the  capillary,  Pi  and  Pz. 
We  will  first  suppose  that  Boyle's  law  holds,  the  flow  taking  place 
isothermally.  For  this  case,  as  we  shall  see,  page  243,  the  vis- 
cosity is  independent  of  the  density.  Let  U,  P,  and  p  represent 
the  mean  velocity,  absolute  pressure,  and  density  at  any  cross- 
section  of  the  tube.  Since  at  any  instant  the  quantity  Q  of  the 
fluid  passing  every  cross-section  is  constant,  we  have  from  Eq.  (4) 
*g£4  dp  *gR*  p  dP*  ,  } 

Q~pu  =^Tp~di  =  -^T^-dT 

But  t  „     ^  is  constant  and  therefore  — jr  =  — — ^ — — 
16?7  P  dl  I 


50  FLUIDITY  AND  PLASTICITY 

and  we  obtain 


or 

*  }  ,     } 


where  Fi  and  F2  are  the  volumes  corresponding  to  pressure 
Pi  and  P2. 

Since  p  =  Pi  —  P2  we  observe  that  'the  ratio  between  the 

P    I    p 

values  of  the  viscosity  calculated  by  Eqs.  (17)  and  (5)  is    *^p 

where  P  may  have  any  value  between  PI  and  P2  depending  upon 
the  value  of  V  which  is  employed.  If  V  be  taken  as 
2Px  2P2     Tr 


Pi+P2 
(17)  becomes  identical  with  Eq.  (5)  and  becomes  unnecessary. 

The  derivation  of  the  law  for  gases  was  made  by  0.  E.  Meyer 
(1866)  and  by  Boussinesq  (1868).  With  Fisher  (1903)  we  may 
regard  the  above  case  where  PF  is  constant  as  extreme,  and  that 
more  generally  we  may  take  PFn  as  constant.  Equation  (16) 
becomes  on  integration 


_        PI    n-p2 

"  ' 


When  n  =  oo  this  becomes  identical  with  Eq.  (4),  for  incom- 
pressible fluids.  When  n  =  1  the  flow  is  isothermal  and  we 
obtain  Eq.  (16a).  Ordinarily  the  value  of  n  will  lie  between  these 
two  extremes,  thus  in  adiabatic  expansion  n  =  CP/CV  =  1.0  to 
1.7,  the  ratio  of  the  specific  heats.  Hence,  it  seems  probable  that 
the  Law  of  Poiseuille  as  given  in  Eq.  (5)  may  be  used,  irrespective 
of  whether  the  fluid  is  compressible  or  not,  but  in  every  case  the 
volume  of  flow  must  be  taken  as 


(18) 


i 

p2  +  ...p2n 

In  the  extreme  case  where  n  =  1,  if  p  is  not  greater  than  P2/10 


AMPLIFICATION  OF  THE  LAW  OF  POI  SEVILLE  51 

V  will  not  differ  from  —  ^-g  —  -   by  much    over   0.2    per  cent. 

This  means  that  working  at  atmospheric  pressure,  with  a  hydro- 
static pressure  of  over  100  cm  of  water,  one  may  take  the  volume 

of  flow  as  -  jr  —  without  any  very  appreciable  error.     It  is 

therefore  extremely  improbable  that  an  appreciable  error  is 
incurred  through  our  lack  of  knowledge  in  regard  to  the  exact 
value  of  n  in  a  given  case.  The  effect  of  the  temperature  upon 
the  viscosity  will  be  discussed  later,  page  246,  as  a  temperature 
correction. 

The  kinetic  energy  of  the  fluid  increases  as  it  passes  along  the 
tube,  but  we  are  interested  only  in  the  total  amount  of  the 
kinetic  energy  as  the  fluid  leaves  the  tube.  This  is  ?rp2.R2723. 
The  total  energy  supplied  in  producing  the  flow  is7rR272(Pi  —Pz)g 
and  the  difference  between  the  two  is  the  energy  converted  into 
heat  7rfl2/2[(Pi  -  Pz)g  -  ?2/22].  The  loss  of  head  in  dynes  per 
cm2  in  imparting  kinetic  energy  to  the  fluid  is  therefore 


With  this  correction,   but  neglecting  the  slipping,   we  obtain 
vg&l  Pi2  -  P22    _  mp2F2 
SVzl         2P2  &rft 

Substituting  V  for  F2  and  remembering  that  p2F2  is  constant, 
Eq.  (19)  becomes  identical  with  the  complete  formula  for  the 
viscosity  as  given  in  Eq.  (17). 

Although  it  is  admitted  that  the  flow  of  compressible  fluids 
is  not  quite  linear,  no  correction  for  this  has  yet  been  attempted. 
However  it  is  certain  that  the  correction  is  negligible  if  p  is  small 
in  comparison  with  P2.  The  correction  for  slipping  in  gases 
plays  an  important  part  in  the  literature.  The  correction  is  the 
same  as  for  incompressible  fluids. 

Turbulent  Flow  in  Gases.  —  The  distinction  between  viscous 
and  turbulent  flow  in  gases  has  been  investigated  by  several 
workers,  among  whom  we  may  mention  particularly  Grindley 
and  Gibson  (1908)  and  Ruckes  (1908).  Ruckes  discovered  that 
the  criterion  for  gases  was  greatly  raised  if  the  capillary  was  blown 
out  into  a  trumpet  shape. 


52 


FLUIDITY  AND  PLASTICITY 


Plastic  Flow  or  the  Fourth  Regime.— When  a  mixture  of  liquids, 
such  as  petroleum,  is  allowed  to  flow  through  a  tube  of  large 
diameter  filled  with  finely  porous  material  like  Fuller's  earth, 
Gilpin1  and  others  have  shown  that  there  is  a  tendency  for  the 
more  volatile,  i.e.  the  more  fluid  substances,  to  pass  through 
the  maze  of  capillaries  first,  leaving  the  more  viscous  substances 
behind.  Naturally  this  effect  is  greatest  when  the  pressure  is 
very  small.  It  is  easy  to  see  that  under  such  conditions  of  flow 
the  fluidity  as  calculated  might  appear  quite  abnormal.  Just 
as  the  fluidity  appears  abnormal  when  the  velocity  exceeds  a 
certain  value  and  we  pass  into  the  second  regime,  so  it  appears 
that  the  fluidity  may  appear  abnormal  when  the  velocity  drops 
below  a  certain  critical  value,  and  we  pass  into  what  may  be 
called  the  "Fourth  Regime." 

With  homogeneous  liquids  or  gases  of  high  fluidity  it  is  diffi- 
cult to  work  at  excessively  low  velocities,  particularly  on 
account  of  the  interference  of  dust  particles.  Very  little  work 
has  been  done  upon  such  substances  having  low  fluidity,  so  that 
for  aught  we  know  now  the  lower  critical  velocity  may  be  ob- 
servable only  in  mixtures. 

Glaser  (1907)  measured  the  viscosity  of  colophonium-turpen- 
tine  mixtures  by  the  transpiration  method  with  the  object  of 
testing  the  law  of  Poiseuille  for  very  viscous  and  plastic  sub- 
stances. With  one  tube  R  =  0.49  cm,  I  =  10.5  cm  he  found 

TABLE  XVIII. — THE  VISCOSITY  OP  AN  85  PER  CENT   COLOPHONIUM — 15 

PER  CENT  TURPENTINE  MIXTURE  AT  11.3°  AND  UNDER  A  CONSTANT 

PRESSURE  OP  2,040  CM  WATER    IN  TUBES  OF 

VARIOUS  DIAMETERS  (GLASER) 


R 

I 

t 

V 

r,  X  107 

V  X  10~8 

1.525 

25.1 

600 

2.28 

4.20 

2.38 

1.019 

15.9 

1,800 

2.30 

4.21 

2.37 

0.746 

16.0 

900 

0.329 

4.25 

2.35 

0.576 

15.1 

18,000 

1.972 

4.22 

2.36 

0.364 

15.8 

46,800 

0.755 

5.22 

1.80 

0.257 

15.2 

43,200 

0.149 

6.59 

1.51 

0.158 

15.1 

173  ,  500 

0.023 

19.90 

0.50 

0.117 

15.4 

3  weeks 

0.000 

00 

0.00 

1  Am.  Chem.  J.,  40,  495  (1908);  44,  251  (1910);  50,  59  (1913). 


AMPLIFICATION  OF  THE  LAW  OF  POISEUILLE 


53 


the  product  of  the  pressure  multiplied  by  the  time  of  efflux  to 
be  constant.  The  velocities  ranged  from  0.00011  to  0.00175 
cm  per  second.  From  these  experiments  Glaser  concluded  that 
"The  velocity  of  efflux  in  this  mixture  is  within  very  wide  limits 
without  influence  upon  the  magnitude  of  the  viscosity."  But 
in  experimenting  with  tubes  of  varying  diameter,  he  obtained 
remarkable  results  a  part  of  which  are  given  in  Table  XVIII. 
We  observe  with  Glaser  that  the  fluidity  rapidly  falls  off 
as  soon  as  the  diameter  falls  below  a  certain  limit.  But  this 
limit  depends  upon  the  fluidity  of  the  mixture,  as  was  proved 


2.5, 


2.0 


1.0 


0.5 


<^xlO' 


7 


0.5 


1.0 


1.5 


Radius  of  Tube 

FIG.  18. — Eighty,  eighty-five  and  ninety  per  cent  mixtures  of  colophonium 
and  turpentine  give  fluidities,  multiplied  by  10~6,  10~8  and  10~10  respectively, 
which  vary  with  the  radius  of  the  tube.  Such  mixtures  are  apparently  plastic 
and  do  not  obey  the  laws  of  viscous  flow. 


by  working  in  the  same  way  with  80  and  90  per  cent  colopho- 
nium mixtures,  the  true  fluidities  of  which  are  approximately 
2  X  10~6  and  2  X  10~10  respectively.  Since  with  such  rapidly 
increasing  values,  the  viscosities  are  inconvenient  to  plot,  we 
have  changed  his  viscosities  to  fluidities.  All  of  his  values  are 
plotted  in  Fig.  18,  using  apparent  fluidities  as  ordinates  and 
radii  as  abscissas.  We  note  that  the  points  lie,  for  the  most 


54  FLUIDITY  AND  PLASTICITY 

part,  on  a  smooth  curve  indicating  that  the  phenomenon  under 
consideration  is  not  one  of  mere  clogging,  as  by  accidental  dust 
particles  in  ordinary  measurements.  The  effect  is  pronounced 
in  a  tube  of  about  0.8  cm  radius  with  a  90  per  cent  colopho- 
nium  mixture,  but  the  effect  is  not  noticeable  in  a  tube  of  0.1 
cm  radius  with  an  80  per  cent  mixture.  It  is  therefore  no 
wonder  if  this  effect  is  not  noticeable  in  ordinary  liquids  which 
are  millions  of  times  yet  more  fluid. 

The  fact  which  seems  to  have  been  overlooked  by  Glaser 
and  is  of  prime  importance  in  explaining  the  phenomenon,  is 
that  the  shearing  stress  and  the  mean  velocity  of  efflux  is  very 
much  less  in  the  smaller  tubes.  Obtaining  the  critical  value  of 
the  radius  for  each  mixture  by  the  graphical  method,  we  have 
calculated  the  mean  velocity  by  means  of  Eq.  (6).  It  is  of  the 
same  order  of  magnitude  in  all  three  cases  being  around  0.000,01 
cm  per  second.  It  seems  probable  that  had  these  experiments 
been  repeated  at  a  very  greatly  different  pressure,  it  would  have 
been  discovered  that  the  viscosity  is  dependent  upon  the  shear- 
ing force  rather  than  upon  the  radius  of  the  tube,  and  the  con- 
clusion that  the  viscosity  is  independent  of  the  velocity  would 
have  been  amended.  It  is  highly  desirable  that  experiments 
be  made  to  establish  this  point. 

It  is  important  to  observe  that  each  mixture  used  by  Glaser 
gave  a  zero  fluidity  when  the  radius  of  the  tube  fell  below  a 
certain  well-defined  limit.  Bingham  and  Durham  (1911) 
have  studied  various  suspensions  of  clay,  graphite  et  cetera  in 
different  liquids  over  a  range  of  temperatures,  using  a  single 
capillary  and  a  nearly  constant  pressure.  As  shown  in  Fig.  19, 
the  fluidity-concentration  curves  are  all  linear  and  at  all  concen- 
trations and  temperatures  they  point  to  a  well-defined  mixture 
with  zero  fluidity,  at  no  great  concentration.  This  mixture 
apparently  sharply  demarcates  viscous  from  plastic  flow,  for 
be  it  noted  that  the  mixture  having  "zero  fluidity"  was  not  a 
hard  solid  mass,  but  rather  of  the  nature  of  a  thin  mud.  In 
the  mixture  of  "zero  fluidity"  it  appears  that  with  the  given 
instrument  all  of  the  pressure  is  required  for  some  other  purpose 
than  to  produce  viscous  flow.  The  amount  of  pressure  used  up 
in  this  way  is  zero  for  the  suspending  medium  alone  but  increases 
in  a  linear  manner  with  the  concentration  of  solid.  If  this  view  is 


AMPLIFICATION  OF  THE  LAW  OF  POI SEVILLE 


55 


correct,  a  part  of  the  pressure  is  used  up  in  producing  plastic 
flow  and  the  rest  in  producing  viscous  flow.     If  higher  pressures 


220 
310 
200 
190 
180 

no 

160 
150 
140 
130 

120 
3) 

*  no' 
l-oo 

90 
80 
10 
60 
50 
40 
30 
20 
10 

°0 

\ 

\ 

\ 

K 

\ 

\ 

\ 

\ 

\ 

K 

\ 

\ 

> 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

go 

^0 

\ 

\ 

^" 

\ 

\ 

\ 

^ 

^ 

\ 

\ 

\ 

^ 

V 

\ 

\ 

\ 

\ 

^ 

\ 

\1 

\ 

\ 

V 

\ 

\ 

v 

\ 

\ 

\  . 

s\ 

\ 

s\ 

V 

\  , 

s\ 

\ 

\ 

^ 

A 

\\ 

\ 

^ 

\N 

N 

s 

^ 

^ 

\ 

" 

®s 

^ 

^ 

\ 

^ 

^ 

\ 

1       2      3      4      5      <b      1      8      9      10      H      12     13      U 

PercerHicige  Volume  of  Earfh 
FIG.  19. — The  fluidity  of  suspension  of  infusorial  earth  in  water. 

were  used  the  zero  of  fluidity  would  be  changed  because  there 
would  be  enough  pressure  to  more  than  overcome  the  plastic 
resistance  in  the  mixture  that  formerly  had  "zero  fluidity." 


56 


FLUIDITY  AND  PLASTICITY 


Further  work  is  therefore  demanded  in  order  that  we  may  clearly 
define  and  separate  the  coefficients  of  plasticity  and  fluidity 
which  are  here  measured  together. 

Surface  Tension  and  Capillarity. — Several  investigators  have 
attempted  to  measure  viscosity  by  means  of  a  capillary  opening 
directly  into  the  air.  Poiseuille  (1846)  found  that  whether  drops 
were  allowed  to  form  on  the  end  of  the  capillary  or  the  end  of  the 
capillary  was  kept  in  contact  with  the  wall  of  the  receiving 
vessel,  he  was  unable  to  obtain  consistent  results.  The  effect 
of  surface  tension  varies  with  the  rate  of  flow,  with  the  tempera- 
ture, and  it  also  depends  upon  the  shape  and  position  of  the  end 
of  the  capillary,  so  that  as  a  whole  the  effects  are  quite  indeter- 
minate. That  the  effects  are  large  and  variable,  may  be  inferred 
from  the  measurements  of  Ronceray  (1911)  with  a  capillary, 
I  =  10.5  cm,  R  =  0.0275,  immersed  under  water  or  opening 
into  the  air,  given  in  Table  XIX. 

TABLE  XIX. — EFFECT  OF  SURFACE  TENSION  ON  THE  FLOW  OF  WATER 
(RONCERAY) 


P  centimeters, 
water 

Time  of  flow  of 
10  ml  in  air  at  17° 

Time  of  flow  im- 
mersed 

Difference 

10 

1,132.0 

1,089.44 

42.6 

20 

559.5 

550.4 

9.1 

30 

373.0 

368.5 

4.5 

40 

280.6 

277.4 

3.2 

50 

224.9 

222.7 

2.2 

60 

187.9 

186.1 

1.8 

70 

161.8 

159.5 

2.3 

Poiseuille  recorded  similar  results.  The  irregularity  is  com- 
pletely removed  by  having  the  end  of  the  capillary  immersed. 
Nevertheless  in  an  apparatus  like  that  used  by  Poiseuille  there 
may  still  be  a  correction  for  capillary  attraction  within  the  bulb 
which  is  considerable  (cf.  p.  66). 

Summary. — From  the  foregoing,  it  appears  that  under  proper 
conditions,  the  only  correction  that  it  is  necessary  to  make 
to  the  simple  Law  of  Poiseuille  is  that  for  the  kinetic  energy 

of  the  fluid  as  it  leaves  the  capillary  ~Lr~-     Other  sources 


AMPLIFICATION  OF  THE  LAW  OF  POI SEVILLE  57 

of  error  such  as  surface  tension,  slipping  at  the  boundary, 
necking  in  of  the  lines  of  flow  at  the  entrance  of  the  capillary, 
eddy  currents  inside  of  the  capillary,  resistance  to  flow  outside 
of  the  capillary,  peculiar  shapes  to  the  ends  of  the  capillary 
affecting  the  magnitude  of  the  kinetic  energy  correction  have 
all  been  considered  in  detail.  They  may  all  be  eliminated 
by  using  long,  narrow  capillaries  with  a  low  velocity  of  flow. 
The  fluidity  of  compressible  fluids  may  be  obtained  by  the 
same  Law  of  Poiseuille  but  the  volume  of  flow  is  approximately 
the  mean  of  the  volume  at  the  entrance  and  at  the  exit  of  the 

.„        Vj.  +  V, 
capillary  — ^ . 

Plastic  solids  in  their  flow  do  not  obey  the  Law  of  Poiseuille 
and  their  study  is  deferred  until  Chapter  VIII.  Many  attempts 
have  been  made  to  measure  the  viscosity  of  soft  solids.  The 
fluidity  of  such  a  substance  is  not  a  constant  quantity  but 
falls  off  rapidly  although  regularly  as  the  radius  of  the  capillary 
falls  below  a  certain  point.  This  is  not  stoppage  of  the  capillary 
of  the  ordinary  sort  due  to  extraneous  particles,  but  rather  a  new 
type  of  flow.  The  terms  fluidity  and  viscosity  will  therefore  be 
avoided  when  referring  to  plastic  substances  in  order  to  avoid 
confusion  and  a  sharp  criterion  given  by  which  a  soft  solid  may  be 
distinguished  from  a  true  fluid,  just  as  Reynolds'  criterion 
enables  one  to  distinguish  between  viscous  and  turbulent  flow. 


CHAPTER  IV 
IS  THE  VISCOSITY  A  DEFINITE  PHYSICAL  QUANTITY? 

So  long  as  the  theory  was  so  imperfectly  worked  out  that 
the  values  for  the  viscosity  of  a  well-defined  substance  like 
water  were  different  when  obtained  with  different  forms  of 
instruments,  it  was  inevitable  that  the  whole  theory  and  practice 
of  viscosity  measurement  should  have  been  called  into  question. 
Among  numerous  researches,  we  may  cite  in  this  connection 
those  of  Traube  (1886),  Wetzstein  (1899)  for  liquids,  Fisher 
(1909)  for  gases,  and  Reiger  (1906)  for  solids.  Since  the  limita- 
tions and  corrections  discussed  in  the  preceding  chapter  have 
evolved  very  gradually,  many  of  these  researches  are  now  of 
historical  interest  only,  and  their  discussion  here  would  be  as 
tedious  as  it  is  unnecessary.  Enough  material  has  already  been 
given  to  prove  that  viscosity  is  an  entirely  definite  property  for 
liquids.  Table  IX  proved  that  tubes  of  quite  diverse  dimen- 
sions give  entirely  harmonious  results.  This  has  been  confirmed 
repeatedly,  especially  by  Jacobson,  1860,  working  with  tubes  of 
considerably  larger  bore.  Not  only  are  the  results  obtained 
with  the  transpiration  method  in  agreement,  among  themselves, 
they  also  agree  with  the  results  from  various  other  methods,  as 
shown  in  Table  XIII. 

Knibbs  (1896)  has  made  a  critical  study  of  the  existing  data 
for  water,  recalculating  and  using  the  corrections  suggested 
in  the  last  chapter.  The  result  was  not  satisfactory.  Many  of 
the  measurements  were  found  to  be  uncertain  and  as  a  result  of  his 
study  Knibbs  doubted  whether  it  was  yet  possible  to  determine 
the  viscosity  of  a  substance  like  water  with  an  error  of  much  less 
than  1  per  cent  from  0  to  50°,  or  5  per  cent  from  50  to  100°. 
During  the  last  20  years  investigations  have  been  carried  out, 
which  give  thoroughly  satisfactory  and  concordant  results,  as  is 
shown  by  Table  II  in  Appendix  D.  The  improvement  is  due 
to  a  happier  disposition  of  apparatus  for  controlling  the  different 
correction  factors. 

58 


IS  THE  VISCOSITY  A  DEFINITE  PHYSICAL  QUANTITY?     59 

Among  gases  air  may  be  regarded  as  the  standard  substance 
as  water  is  among  liquids.  And  if  we  compare  the  numerous 
values  for  the  viscosity  of  air  obtained  prior  to  20  years  ago,  the 
result  is  discouraging  and  has  been  often  commented  upon. 
These  values  are  given  for  0°  in  Table  XX. 

TABLE  XX.— FLUIDITY  OF  AIR  AT  0°C. 


Method 

*> 

9 

Observer 

Transpiration  

5,942 

Graham  (1846) 

Oscillating  disks  



5,325 

Maxwell  (1866) 

Oscillating  disks  

5,814 

Meyer  and  Springmuhl  (1873) 

Oscillating  disks  

5,590 

Puluj  (1874) 

Oscillating  disks  

5,556 

Puluj  (1874) 

Transpiration  

5,854 

Obermayer  (1875) 

Oscillating  disks  

5,489 

Puluj  (1876) 

Transpiration  

5,951 

Obermayer  (1876) 

Transpiration  

5,650 

E.  Wiedemann  (1876) 

Transpiration  

5,988 



Obermayer  (1876) 

Transpiration  

5,952 

Obermayer  (1876) 

Transpiration  

5,848 

O.  Meyer  (1877) 

Transpiration  

5,882 

O.  Meyer  (1877) 

Transpiration  

5,747 

O.  Meyer  (1877) 

Oscillating  disk  

5,714 

Puluj  (1878) 

Transpiration  

5,650 

Hoffman  (1884) 

Oscillating  disk  

5,955 

Schumann  (1884) 

Oscillating  disk  

5,838 

Schneebeli  (1885) 

Oscillating  cylinder.  .  .  . 

5,831 

Tomlinson  (1886) 

Transpiration 

5,770 

Breitenbach  (1899) 

Oscillating  cylinder  .... 

5,659 

F.  Reynolds  (1904) 

Transpiration  

5,761 

Tanzler  (1906) 

The  transpiration  method  appears  to  give  higher  values  for  the 
fluidity  than  are  obtained  by  the  other  methods  but  the  results 
are  not  very  consistent  among  themselves.  However  the  follow- 
ing table  of  recent  values  for  the  viscosity  of  air  at  15°  is  very 
satisfactory. 

We  have  the  authority  of  Fisher  (1909)  page  150,  for  the  state- 
ment that  "No  experimenter  has  made  the  attempt  to  apply 
a  correction  to  his  measured  pressures  to  allow  for  the  kinetic 
energy  of  the  emerging  gas."  It  appears  probable  that  the  exist- 


60  FLUIDITY  AND  PLASTICITY 

TABLE  XXI.— FLUIDITY  OF  Am  AT  15° 
Method  <f  Observer 


Transpiration  

5,507 

Breitenbach  (1899) 

Transpiration 

5  502 

Schultze  (1901) 

Transpiration 

5  528 

Markowski  (1904) 

Transpiration  .  . 

5  502 

Schmitt  (1909) 

Transpiration  

5.531 

Knudsen  (1909) 

ing  data  might  be  improved  by  a  critical  study  for  the  purpose 
of  making  the  needed  corrections.  It  is  a  curious  fact  that  the 
kinetic  energy  correction  has  been  so  little  understood  and  appre- 
ciated. Even  in  the  case  of  liquids  it  is  very  commonly  neglected 
although  it  may  amount  to  several  per  cent  of  the  viscosity  to 
be  measured.  It  is  sometimes  stated  that  no  kinetic  energy 
correction  is  necessary  when  liquids  flow  through  a  capillary  from 
one  reservoir  to  another  and  not  into  the  air.  The  Ostwald 
viscometer  (c/.  p.  75),  is  used  more  than  any  other  but  it  appears 
that  no  kinetic  energy  correction  is  ever  applied.  It  is  true  that 
the  instrument  is  used  for  relative  measurements  only,  but  this 
fact  does  not  cause  this  correction  to  be  without  effect  in  the  cal- 
culation, for  the  reason  that  the  kinetic  energy  correction  is  not 
proportional  to  the  viscosity. 

There  may  be  those  who  would  maintain  that  the  correction 
for  kinetic  energy,  as  given  above,  is  not  the  correct  one  to  apply 
to  gases,  c/.  Fisher  (1909).  But  that  a  correction  is  unnecessary 
cannot  be  maintained  even  in  the  case  of  gases,  in  view  of  Hoff- 
mann's work  on  the  interrupted  flow  of  gases.  One  of  his  capil- 
laries was  cut  into  28  pieces  without  loss.  In  flowing  through  the 
interrupted  capillary,  the  kinetic  energy  correction  would  be 
increased  twenty-eight  fold,  as  has  been  already  indicated  on  page 
28.  As  a  matter  of  fact  the  time  of  flow  was  considerably 
greater  in  the  interrupted  capillary,  proving  the  importance  of  the 
correction.  It  would  be  particularly  interesting  to  see  whether 
an  experimental  verification  of  the  correction  would  be  obtained 
by  an  intensive  study  of  the  data  of  Hoffmann  (1884). 

The  term  "specific  viscosity"  has  been  very  largely  used  and 
may  here  receive  brief  comment.  Water  at  0°  has  been  taken 
as  a  standard  with  a  specific  viscosity  of  100,  but  water  at  25° 


IS  THE  VISCOSITY  A  DEFINITE  PHYSICAL  QUANTITY?     61 

has  also  been  taken  as  the  standard  and  equal  to  unity.  Still 
other  standards  have  been  employed.  The  principal  advantages 
in  this  form  of  expression  are  the  saving  of  labor  in  calculation, 
the  avoidance  of  inconveniently  small  fractions,  and  the  use  of  a 
common  liquid  as  standard.  This  advantage  however  is  more 
than  offset  by  the  disadvantages.  The  proper  corrections  are 
never  applied  to  specific  viscosities  and  consequently  the  values 
are  not  really  comparable  among  themselves.  Certainly  they  are 
inconvenient  to  use  for  reference  and  for  comparison  with  vis- 
cosities calculated  in  other  ways.  The  time  necessary  for  the 
preparation  of  the  substances  is  nearly  always  great  enough  to 
justify  the  inconsiderable  expenditure  of  time  necessary  for  the 
proper  reduction  of  the  data  to  absolute  units. 

Of  course  much  depends  upon  the  disposition  of  the  apparatus 
used  in  the  measurement.  Some  forms  of  apparatus  will  not 
permit  accurate  estimations  of  the  viscosity  to  be  made.  But 
given  an  apparatus  which  is  well-suited  for  precise  measurements, 
the  time  required  for  making  a  measurement  is  no  greater,  and 
may  be  much  less,  than  in  the  less  accurate  forms  of  apparatus. 

The  Centipoise. — In  expressing  viscosities,  it  is  possible  to 
secure  simultaneously  the  advantage  of  expression  in  absolute 
units  with  the  advantages  of  viscosities  relative  to  some  common 
substance  as  standard.  It  is  proposed  to  name  the  absolute  unit 
of  viscosity  after  Poiseuille  the  "poise,"  and  consequently  the 
submultiple  of  this  unit  which  is  one-hundredth  as  large  the 
"  centipoise  "  (cp).  It  so  happens  that  one  centipoise  is  almost  ex- 
actly the  viscosity  of  water  at  20°C,  hence  absolute  viscosities 
expressed  in  centiposes  are  also  specific  viscosities  referred  to  water 
20°C  as  standard.  To  be  sure  the  viscosity  of  water  is  not  ex- 
actly one  centipoise  at  20°C  but  it  is  1.005  which  is  unity  within 
the  limits  of  possible  experimental  error  in  ordinary  measurement 
(c/.  Appendix  D,  Table  II). 

All  fluidities  are  expressed  in  absolute  units,  water  at  20° 
having  of  course  a  fluidity  of  100  units. 


CHAPTER  V 
THE  VISCOMETER 

A  very  full  discussion  has  been  given  of  the  theory  of  the  trans- 
piration method.  Much  matter  of  great  historical  interest  in 
regard  to  the  various  other  methods  has  been  passed  over.  This 
has  been  done  in  order  to  present  the  matter  which  will  be  of 
greatest  use  to  the  worker.  At  present  the  transpiration  method 
is  by  all  odds  the  most  important,  this  superiority  being  based 
upon  the  following  advantages:  (1)  It  is  susceptible  of  simple 
mathematical  treatment.  (2)  It  is  rapid.  (3)  Only  a  small 
amount  of  fluid  is  required.  (4)  It  can  be  used  under  the  widest 
variety  of  conditions  as  regards  temperature,  pressure  et  cetera. 
(5)  The  preliminary  measurements  and  adjustments  are  not 
difficult  to  make.  (6)  Finally,  it  has  been  tested  out  most 
thoroughly  and  found  to  be  capable  of  the  highest  degree  of 
precision. 

For  certain  purposes  other  methods  must  apparently  continue 
to  be  used.  Thus  the  pendulum  method  seems  best  suited  for 
investigating  superficial  viscosity  and  the  viscosity  of  solids  like 
steel.  The  fall  method  is  of  great  use  also  for  certain  purposes. 

It  is  quite  impracticable  to  discuss  here  the  almost  innumerable 
forms  of  instruments  which  have  been  suggested  for  use,  so  we 
propose  to  consider  briefly  some  of  the  instruments  which  have 
shown  the  greatest  advance  toward  meeting  the  conditions  of 
an  ideal  disposition  of  apparatus. 

Naturally  all  transpiration  instruments  are  based  upon  that 
of  Poiseuille;  but  for  general  purposes,  his  instrument  was  defi- 
cient, since  the  capillary  terminated  directly  into  the  bath  and 
hence  the  apparatus  had  to  be  refilled  after  each  measurement. 
This  difficulty  was  overcome  in  two  forms  of  apparatus  designed 
by  Pribram  and  Handl  (1880),  which  consisted  of  a  capillary  placed 
between  two  tubes  of  larger  bore.  In  the  better  form  shown  in 
Fig.  20,  the  two  tubes  are  vertical,  the  capillary  being  bent. 
The  advantage  of  this  arrangement  is  immediately  apparent 
62 


THE  V1SCOMETER 


63 


because  as  soon  as  a  measurement  has  been  made  in  one  direction, 
the  apparatus  is  ready  for  an  observation  in  the  opposite  direc- 
tion. With  this  apparatus  Pribram  and  Handl  made  very 
numerous  observations  on  organic  liquids  over  a  range  of  tempera- 
ture. They  used  a  constant  pressure  head. 

The  apparatus  of  Bruckner  (1891)  marked  another  step  in 
advance.  He  used  a  horizontal  capillary  K, 
Fig.  21,  connected  to  the  two  limbs  of  the 
apparatus  by  means  of  short  pieces  of  rubber 
tubing.  Two  reservoirs  R  and  R'  served  for 
the  deposition  of  any  dust  particles  that  might 
have  found  their  way  into  the  liquid.  The 
volumes  of  flow  were  accurately  measured  by 
the  volumes  of  the  bulbs  V  and  V,  the  tubes 
leading  from  these  bulbs  being  constricted 


FIG.  20. — Viscometer  of 
Pribram  and  Handl. 


FIG.  21. — Viscometer  of  Bruckner. 


in  order  to  give  a  sharp  reading.     Either  limb  could  be  turned 
to  pressure  at  H  or  H' ,  or  to  air  at  Hi  or  H'\. 

In  the  study  of  organic  liquids  rubber  connections  become 
objectionable,  hence  Thorpe  and  Rodger  (1894)  in  their  monu- 
mental work  on  the  relation  between  the  viscosity  of  liquids  and 
their  chemical  nature,  employed  an  instrument,  Fig.  22,  similar 
to  that  of  Bruckner  except  that  the  capillary  was  placed  inside 
of  a  wider  tube  which  was  itself  subsequently  sealed  to  the  two 
limbs  of  the  viscometer.  The  middle  of  this  tube  was  heated  at 


64 


FLUIDITY  AND  PLASTICITY 


its  middle  point  until  it  attached  itself  to  the  capillary  all  of  the 
way  around,  the  greatest  care  being  taken  not  to  decrease  the 
diameter  of  the  capillary  or  change  it  in  any  way. 

Another  marked  improvement  was  the  introduction  of  the 
traps  Tl  and  T2,  Fig.  22,  for  the  purpose  of  easily  adjusting  the 
total  volume  of  liquid  within  the  instrument,  which  they  denoted 
as  the  "  working  volume"  to  distinguish  it  from 
the  volume  of  efflux  7.  By  keeping  the  work- 
ing volume  constant,  the  correction  for  the 
hydrostatic  pressure  within  the  instrument  is 
greatly  simplified. 

Unfortunately  Thorpe  and  Rodger's  instru- 
ment has  not  come  into  general  use.  This  is 
probably  due  to  the  following  disadvantages: 
The  sealing  of  the  wide  tube  to  the  middle  of 
the  capillary  is  difficult  to  accomplish;  and 
according  to  Knibbs  (1895)  and  Blanchard 
(1913)  one  cannot  be  sure  that  the  bore  of  the 
capillary  has  not  been  altered  in  spite  of  the 
utmost  precaution.  It  is  practicable  to  get 
the  dimensions  of  the  capillary  and  the  other 
constants  of  the  instrument  only  after  the 
sealing  has  been  completed.  To  get  them  then 
is  a  matter  of  some  difficulty  if  not  uncertainty. 
The  instrument  is  difficult  to  clean  and  dry 
on  account  of  the  narrow  spaces  between  the 
capillary  and  the  wider  tube  at  either  side  of 
the  constriction  R.  At  the  same  time  the 
instrument  is  rather  fragile.  These  difficulties 
may  all  be  overcome  by  using  ground  glass 
joints  between  the  capillary  and  the  two 
limbs.  Over  the  ground-glass  joints  rubber  tubing  may  be 
stretched  and  tied  and  thus  any  danger  of  a  leak  guarded 
against.  The  absence  of  a  leak  can  be  proven  very  easily  at  any 
time  by  simply  testing  the  working  volume.  By  having  goocl 
ground-glass  joints  there  can  be  no  change  in  volume  due  to  the 
change  in  the  expansion  of  the  rubber  under  the  changing  head 
and  there  can  be  no  solvent  action  except  that  on  the  glass  itself. 
On  the  other  hand,  by  planing  the  ends  of  the  capillary  off  at 


FIG.  22. — Viscom- 
eter  of  Thorpe  and 
Rodger. 


THE  VJSCOMETER  65 

right  angles  to  the  axis  of  the  tube,  the  dimensions  of  the  capillary 
may  be  most  accurately  ascertained.  There  is  the  further 
advantage  that  other  capillaries  may  be  used  without  changing 
the  other  constants  of  the  apparatus.  It  is  perhaps  unneces- 
sary to  add  that  this  instrument  must  be  modified  to  make  it 
suitable  for  the  measurement  of  gases. 

The  Most  Suitable  Dimensions  for  a  Viscometer.  —  As  we 
shall  see  in  Part  II,  it  is  more  convenient  to  compare  fluidities 
than  viscosities.  Combining  Eqs.  (8)  and  (13)  we  obtain  for 
the  general  formula  for  the  fluidity  of  all  fluids 

*' 


where  R  is  the  radius  and  I  the  length  of  the  capillary  in  cen- 
timeters, and  V  is  the  volume  in  cubic  centimeters,  all  being 
reduced  to  the  same  temperature.  No  correction  is  necessary  for 
changes  in  these  dimensions  later  since  the  changes  just  neutralize 
each  other  as  can  be  proved  by  introducing  the  coefficient 
of  expansion  into  the  above  formula.  The  pressure  p,  expressed 
in  grams  per  square  centimeter,  is  the  difference  between  the 
absolute  pressures  at  the  level  of  the  capillary  at  the  entrance,  PI 
and  exit,  P%.  The  time  of  flow  in  seconds  is  t,  for  the  fluid  whose 
density  is  p  at  the  temperature  of  observation  T.  ir  =  3.1416, 
g  is  the  acceleration  due  to  gravitation,  m  =  1.12,  X  is  the 
coefficient  of  slipping,  which  is  negligible  for  all  liquids  but  is 
of  importance  in  rarefied  gases,  cf.  page  244.  In  the  measure- 
ment of  the  fluidity  of  gases  the  volume  of  efflux  must  be  cal- 
culated according  to  the  formula  of  page  50. 


V  -  l     V    -  2     V 

~P1  +  P2~P1  +  P2  ' 

where  Fi  is  the  volume  of  flow  as  measured  before  flow  under 
pressure  PI  and  F2  is  the  volume  after  expansion  to  the  pressure 
P2. 

By  the  proper  choice  of  the  dimensions  of  the  apparatus 
an  accuracy  of  one-tenth  of  1  per  cent  may  probably  be  attained. 
With  a  stop-watch  reading  to  0.2  sec.  the  time  of  flow  may  be 
made  as  small  as  200  sec.  The  volume  of  flow  should  be  small 
for  the  following  reasons.  (1)  The  kinetic  energy  correction 
mpF2  in  Eq.  (20)  should  be  kept  from  becoming  inconveniently 


66  FLUIDITY  AND  PLASTICITY 

large.  (2)  The  time  of  flow  should  be  small  for  the  sake  of 
economy  and  also  that  the  temperature  may  be  more  readily  kept 
constant  during  the  time  of  flow.  (3)  Small  masses  of  fluid  come 
to  the  temperature  of  the  bath  more  quickly,  and  (4)  there  is 
an  economy  in  material,  which  is  sometimes  very  important. 
The  minimum  of  flow  is  determined  by  our  ability  to  read  the 
volume  with  the  desired  accuracy.  This  in  turn  is  determined  by 
the  diameter  of  the  constricted  portions  of  the  instrument  above 
and  below  the  measured  volume  V.  If,  however,  the  constricted 
parts  are  of  very  small  bore,  the  capillary  action  becomes  dis- 
turbing. Very  viscous  liquids  will  not  drain  out  properly  and 
they  may  even  form  a  meniscus  across  the  capillary  which  will 
prevent  the  transmission  of  the  pressure  and  will  render  the 
results  quite  valueless.  It  may  be  remarked  that  the  troubles 
due  to  bad  drainage  may  be  minimized  by  having  the  drainage 
surfaces  everywhere  as  nearly  vertical  as  possible.  In  other 
words,  the  change  from  constricted  portion  to  the  tube  of  large 
diameter  should  be  made  gradually.  If  the  constricted  part  of 
the  instrument  has  an  inside  diameter  of  0.25  cm  we  believe 
that  the  capillary  correction  will  not  cause  trouble.  The 
volume  per  centimeter  of  the  constricted  tube  is  then  0.05  ml 
and  if  we  assume  that  the  meniscus  can  be  read  to  0.01  cm  as  it 
passes  a  mark  on  the  tube,  it  is  only  necessary  to  have  a  volume 
of  0.5  ml  to  obtain  the  desired  accuracy.  To  provide  a  margin 
of  safety  in  the  construction  and  use  of  the  apparatus  we  select 
about  3  ml  as  the  minimum. 

To  detect  any  error  due  to  faulty  drainage,  it  is  only  necessary 
to  test  the  flow  of  the  most  viscous  liquid  to  be  measured  using 
very  different  rates  of  transpiration  by  varying  the  pressure. 
Lack  of  perfect  drainage  will  be  made  evident,  by  the  substance 
appearing  to  be  more  viscous  at  the  lower  rate  of  flow.  Natur- 
ally the  more  viscous  liquids  must  be  allowed  to  flow  slowly 
enough  so  that  the  drainage  will  appear  to  be  perfect.  If  in  the 
instrument  depicted  in  Fig.  23  the  flow  were  to  begin  with  the 
upper  meniscus  at  the  point  marked  3,  it  would  be  necessary  for 
all  of  the  liquid  of  the  measured  volume  V  to  have  drained  out  at 
the  expiration  of  the  time  t.  This  is  not  necessary,  however,  if 
the  flow  begins  at  some  point  considerably  higher  up,  as  for  exam- 
ple in  the  neighborhood  of  the  trap-opening  F,  for  then  a  certain 


THE  VISCOMETER 


r>7 


FIG.   23. — Viscometer  for  absolute  measurements. 


68  FLUIDITY  AND  PLASTICITY 

amount  of  liquid  may  flow  into  V  from  E  after  the  record  of  the 
time  has  begun,  and  this  will  tend  to  offset  the  effect  of  any  liquid 
left  in  V  at  the  end  of  the  time  of  flow.  To  make  these  amounts 
as  nearly  equal  as  possible,  the  lower  part  of  E  should  be  exactly 
similar  in  shape  to  the  lower  part  of  V. 

The  pressure  should  be  variable  at  will  so  that  the  time  of 
flow  may  be  kept  reasonably  constant.  For  gases,  high  pressures 
are  as  unnecessary  as  they  are  undesirable.  For  incompressible 
fluids,  there  need  be  no  upper  limit  set  to  the  pressure.  A  pres- 
sure of  50  g  per  square  centimeter  can  easily  be  read  to  0.1  per 
cent  on  a  water  manometer,  and  the  various  pressure  correc- 
tions— to  be  discussed — may  be  ascertained  well  within  this 
limit,  hence  this  may  be  taken  as  a  lower  limit. 

The  measurement  of  the  radius  of  the  capillary  offers  the  great- 
est difficulty  in  viscosity  measurement  by  this  method.  Since 
the  flow  is  proportional  to  the  fourth  power  of  the  radius,  any 
error  in  this  measurement  is  multiplied  four  times.  Careful 
weighing  of  the  quantity  of  mercury  required  to  fill  the  tube  is 
perhaps  the  best  means  for  obtaining  the  mean  radius,  R  = 
•^(W/irpl) ;  but  for  a  capillary  such  as  that  used  by  Thorpe  and 
Rodger,  I  =  4.9+  cm  R  =  0.0082+  cm,  the  weight  of  the  mer- 
cury is  only  about  0.013  g  so  that  the  desired  accuracy  is  diffi- 
cult to  obtain  with  the  ordinary  balance.  If  the  radius  is 
increased,  the  time  of  flow  may  be  kept  constant  by  increasing 
the  length  so  that  the  ratio  l/R*  is  constant.  Fortunately  both 
of  these  changes  tend  to  increase  the  volume  of  the  capillary. 
At  the  same  time  the  increase  in  length  diminishes  the  effect 
of  any  possible  alteration  in  the  stream  lines  near  the  ends;  and 
the  increase  in  the  radius  diminishes  the  possible  effect  of  slip- 
ping and  probably  also  the  effect  of  dust  particles. 

The  formula  (20)  applies  only  to  a  capillary  which  has  the 
form  of  a  true  cylinder,  but  usually  the  capillary  is  elliptical  and 
it  may  at  the  same  time  be  conical.  To  determine  the  conicity, 
the  tube  must  be  calibrated  with  a  mercury  thread.  To  deter- 
mine the  ratio  of  the  axes,  the  micrometer  microscope  should 
be  used.  In  using  the  micrometer  microscope  it  is  somewhat 
difficult  to  see  the  exact  circumference  to  be  measured,  owing  to 
various  causes.  Poiseuille  found  it  best  to  grind  off  and  polish 
the  end  of  the  tube  and  then  attach  a  cover-slip  to  this  end  by 


THE  VISCOMETER  69 

means  of  Canada  balsam  which  is  warmed  slightly  until  it  fills 
the  end  of  the  capillary. 

If  the  capillary  is  elliptical,  R*  in  Eq.  (20)  must,  according 
to  Riicker  (cf.  Thorpe  and  Rodger  (1893)),  be  given  the  value 

2B3CS 
P2  _i_  rz  wnere  B  and  C  are  the  major  and  minor  axes  of  the  ellip- 

.D      "~T~  0 

tical  cross-section.     If  the  capillary  is  the  frustrum  of  a  circular 
cone,  Knibbs  has  shown  that  R*  must  be  replaced  by 


where  R  and  Rz  are  the  radii  of  the  two  ends.  If  the  capillary 
is  at  the  same  time  elliptical,  R*  becomes 

3R3SR*S  (1  -  e2)3 

R32  +  R3Ri  +  RS  '     1  +  ez 

where  R3  and  #4  represent  the  arithmetical  means  of  the  major 

r>  _   ri 

and  minor  radii  at  their  respective  ends,  and  e  =  p   .   n  where 

t>  ~\~  0 

B  and  C  represent  the  mean  semi-axes.  Knibbs  has  also  con- 
sidered the  corrections  necessary  for  other  peculiarities  in  the 
bore  of  the  tube  which  need  not  be  considered  here. 

There  is  no  special  advantage  in  using  a  variety  of  viscometers 
for  liquids  of  not  very  different  fluidity.  For  liquids  below  the 
boiling-point  the  fluidity  never  exceeds  about  500.  Assuming 
this  value  as  the  maximum  the  lengths  necessary  for  a  capillary 
of  a  given  radius  have  been  calculated  by  means  of  Eq.  (5)  and 
plotted  curve  A  in  Fig.  24.  It  is  not  always  possible  to  obtain 
a  capillary  of  an  exactly  specified  radius,  but  with  one  having  an 
approximately  satisfactory  radius,  the  necessary  length  can  be 
read  off  from  the  curve.  For  gases  the  maximum  fluidity  must 
be  taken  as  10,000.  If  only  very  viscous  liquids  are  to  be  meas- 
ured the  maximum  may  be  taken  as  less  than  500,  curve  B  or  C. 
(cf.  also  Appendix  A,  Table  IX.) 

Construction  and  Calibration  of  Apparatus.  —  A  point  of  great 
importance  in  the  construction  of  the  viscometer  is  to  have  the 
volume  V  (1)  as  nearly  equal  to  that  of  V  as  possible,  (2) 
similar  to  it  in  shape,  and  (3)  at  the  same  height  from  the  hori- 
zontal capillary.  This  construction  greatly  facilitates  the  esti- 
mation of  the  correction  for  hydrostatic  pressure,  within  the 


70 


FLUIDITY  AND  PLASTICITY 


instrument.  Finally  the  small  bulbs  C,  E,  'C',  and  Ef  should 
have  nearly  the  same  volume.  By  having  the  surfaces  nowhere 
depart  greatly  from  the  vertical,  the  drainage  is  improved.  It 
is  impracticable  however  to  use  long,  cylindrical  bulbs,  since 
then  the  true  average  pressure,  due  to  the  hydrostatic  head  within 
the  instrument,  becomes  awkward  to  determine.  (C/.  Appendix 
A,  page  297.)  The  best  form  for  the  bulbs  V  and  V  is  therefore 
obtained  by  making  them  so  that  each  resembles  as  much  as 


,f 


JS 


006   .009     .010      .Oil     .012     0.15      0.14     0.15    .016     .017  .  .018     .019      .020    .021 

Radius  in  cm. 

FIG.  24. — Chart  for  use  of  instrument  maker  in  selecting  capillary  for  vis- 
cometer,  knowing  the  approximate  radius  of  the  capillary  and  the  maximum 
fluidity  to  be  measured,  the  length  to  be  used  may  be  read  off.  V  =3ml  t  =200 
sec.,  p  =50  g  per  cm2. 

possible  a  pair  of  hollow  cones,  placed  base  to  base  as  shown  in 
Fig.  23. 

The  marks  at  1  and  3  are  so  placed  that  the  volume  from  3  to 
F  is  exactly  equal  to  that  from  1  to  2'.  If  the  two  limbs  of  the 
apparatus  are  similar  there  will  be  no  correction  for  capillarity. 
Poiseuille  has  given  a  method  for  estimating  this  correction  when 
that  is  necessary.  The  volumes  V  and  V  may  be  easily  deter- 
mined by  the  weight  of  volumes  of  mercury. 

The  appearance  of  the  complete  apparatus  used  by  Thorpe 
and  Rodger  is  shown  in  Fig.  25.  The  viscometer  is  shown  in  the 
bath  B  which  has  transparent  sides.  Water  in  the  vessel  R 
exerts  pressure  upon  the  air  in  the  large  reservoir  L.  The  gas 


THE  VISCOMETER 


71 


FIG.  25. — Complete  viscometer  apparatus  of  Thorpe  and  Rodger. 


72 


FLUIDITY  AND  PLASTICITY 


is  dried  by  passing  over  sulfuric  acid  in  a  smaller  bottle  Mt 
whence  tubes  lead  to  the  three-way  stop  cocks  Z  and  Z'  and 
thence  to  the  two  limbs  of  the  viscometer.  The  pressure  is 
measured  on  the  water  manometer  D.  The  bath  is  stirred  by 
means  of  a  motor  connected  with  the  mechanism  shown  at  E. 
Since  the  fluidity  of  a  substance  like  water  changes  from  1  to  3 
per  cent  with  a  change  of  1°  in  the  temperature,  it  is  necessary 
that  the  temperature  be  controlled  to  a  few  hundredths  of  a 
degree.  Since  they  were  working 
over  a  wide  range  of  temperature, 
Thorpe  and  Rodger  controlled  the 
temperature  by  hand. 

A  word  may  be  added  here  in 
regard  to  stop-watches.  The  com- 
mon form  of  stop-watch  in  which 
the  whole  mechanism  starts  or  stops 
simultaneously  with  the  time  record 
may  not  give  consistent  results,  even 
though  it  appears  to  neither  gain 
nor  lose  during  a  long  period  of  time. 
This  is  the  fault  of  the  mechanism. 
The  watches  whose  movements  con- 
tmue,  whether  the  time  is  being 
recorded  or  not,  seem  to  be  freer 
from  this  defect. 

The  Measurement.  —  In  preparing  substances  for  measurement 
as  well  as  in  cleaning  and  drying  the  instrument,  many  investi- 
gators have  strongly  emphasized  the  importance  of  avoiding  the 
presence  of  dust  particles.  Both  Poiseuille  and  Thorpe  and 
Rodger  took  elaborate  precaution  in  this  regard.  Figure  26 
shows  the  apparatus  used  by  the  latter  for  distilling  pure  liquids. 
It  has  the  advantage  of  allowing  a  good  determination  of  the 
boiling-point  to  be  made  while  the  liquid  is  being  fractionated. 
To  avoid  contamination  by  dust  and  moisture  in  filling  the  vis- 
cometer, Thorpe  and  Rodger  used  a  special  apparatus,  Fig.  27. 
The  liquid  was  placed  in  the  bottle  H  and  forced  over  into  the 
right  limb  of  the  viscometer  M  by  means  of  the  pressure  of  a 
mercury  head  A.  The  viscometer  was  held  in  a  frame  and 
supported  on  the  vertical  rod  by  means  of  the  setscrew  N. 


no.  26.-ApParatu8  of 
Thorpe  and  Rodger  for  obtain- 
ing  dust-free  liquid. 


THE  VISCOMETER 


73 


The  left  limb  of  the  viscometer  was  evacuated  by  means  of 
the  mercury  head  Q  in  order  to  draw  the  liquid  through  the 
capillary. 

Having  run  in  a  little  more  than  the  required  amount  of 
liquid,  the  viscometer  and  frame  were  placed  in  the  bath  B  of 
Fig.  25  and  the  limbs  of  the  viscometer  were  connected  to  the 
pressure  outlets  on  either  side. 
With  the  temperature  main- 
tained constant  at  the  lowest 
point  at  which  measurements 
were  desired,  the  cock  Zf  (or  Z) 
was  turned  to  air  and  the  cock  Z 
(or  Z'}  to  pressure.  As  the  liquid 
rose  in  the  left  limb,  it  finally 
overran  into  the  trap  T,'  Fig.  22. 
At  the  instant  that  the  meniscus 
in  the  right  limb  reached  the 
point  k2,  the  cock  Z  was  turned 
to  air.  Thus  the  working  volume 
was  adjusted. 

A  measurement  of  the  fluidity 
is  made  by  turning  the  cock  Z' 
to  pressure  and  immediately  read- 
ing the  pressure  on  the  manom- 
eter as  well  as  the  temperature  of 
the  manometer,  while  the  liquid 
is  flowing  out  of  the  bulb  V.  As 
the  meniscus  passes  the  point 
m'  the  time  recorded  is  begun. 
Keeping  the  temperature  constant 
the  time  is  taken  as  the  meniscus 

passes    the    point    m2.    The    pres-     FIG.  27. — Filling   device   of   Thorpe 

sure  is  then  read  as  before,  and 


and  Rodger. 


before  the  meniscus  reaches  the  point  k'  the  left  limb  is  again 
turned  to  air.  The  apparatus  is  then  ready  for  a  duplicate 
observation  in  the  opposite  direction. 

The  Calculation. — The  corrections  to  the  time  and  temperature 
are  not  peculiar  to  viscosity  measurements  and  need  no  special 
comment.  In  obtaining  the  pressure,  several  corrections  must 


74  FLUIDITY  AND  PLASTICITY 

be  made.  (1)  The  pressure  on  the  manometer  must  be  calculated 
to  grams  per  square  centimeter  from  the  known  height  of  the 
liquid  and  its  specific  gravity  at  the  temperature  observed.  A 
correction  to  the  observed  height  of  the  liquid  is  avoided  by 
having  the  long  limb  of  the  manometer  doubly  bent  at  its  middle 
point  so  that  the  upper  half  is  vertical  and  in  the  same  straight 
line  with  the  lower  limb  of  the  manometer.  The  levels  on  both 
limbs  may  then  be  read  on  the  same  scale,  which  may  con- 
veniently consist  of  a  steel  tape  mounted  on  a  strip  of  plate-glass 
mirror  placed  vertically.  Similarly  a  correction  for  capillary 
action  may  be  avoided  if  the  bore  of  the  manometer  is  large 
enough  so  that  it  may  be  assumed  to  be  uniform.  (2)  The  pres- 
sure must  be  corrected  for  the  weight  of  the  air  displaced  by  the 
liquid  in  the  manometer.  (3)  Unless  the  surface  of  the  liquid  in 
the  lower  limb  of  the  manometer  is  at  the  same  height  as  the 
average  level  of  the  liquid  in  the  viscometer,  a  correction  must  be 
made  for  the  greater  density  of  this  enclosed  air,  than  of  the 
outside  air  which  is  not  under  pressure.  (4)  Finally  a  correction 
must  be  made  for  the  average  resultant  hydrostatic  head  of  the 
liquid  within  the  viscometer.  If  the  two  volumes  V  and  V 
in  Fig.  23  are  exactly  equal  in  volume,  similar  in  shape,  and 
at  the  same  elevation  above  the  capillary,  when  the  viscometer  is 
in  position,  in  the  bath,  it  is  evident  that  the  gain  in  head  during 
the  first  half  of  the  flow  will  be  exactly  neutralized  by  the  loss 
in  head  during  the  last  half  of  the  flow.  Since  this  cannot  be 
exactly  realized,  a  correction  may  be  made  as  follows :  Duplicate 
observations  in  reverse  directions  are  made  upon  a  liquid  of 
known  density  and  viscosity  at  a  constant  temperature  and 
pressure.  Let  t\  be  the  time  of  flow  from  left  to  right  and  tz  the 
corresponding  time  from  right  to  left.  Let  p0  be  the  pressure  as 
corrected,  except  for  the  average  resultant  head  of  liquid  in  the 
viscometer.  Suppose  this  latter  correction  to  amount  to  x  cm 
of  the  liquid  as  the  liquid  flows  from  left  to  right.  In  this  case 
the  total  pressure  becomes  equal  to  p0  +  px  and  when  the 
liquid  flows  from  right  to  left,  it  becomes  equal  to  p0  —  px. 
Since  Eq.  (8)  when  used  for  a  given  viscometer  may  be  written 
in  the  form 

r,  =  Cpt  -  C'p/t  (22) 


THE  VISCOMETER 


75 


where  C  and  C'  are  constants,  which  can  be  calculated,  we  obtain 

Po  +  PX=  *  + 


whence, 


Po  — 


_ 

2CPU 


Cti 

r,  +  C'p/i* 

Ctz 


^.Ti   n 

2CU2       *22J 


In  subsequent  calculations  it  is  necessary  to  know  the  specific 
gravity  of  the  liquid  whose  viscosity  is  desired,  in  order  to  make 
the  necessary  pressure  correction  and  in  order  to  make  the  kinetic 
energy  correction,  but  it  is  to  be  noted  that  if  the 
instrument  has  been  constructed  with  that  end  in 
view,  these  corrections  will  both  be  small,  and  there- 
fore the  specific  gravity  need  be  only  approximately 
known,  which  is  a  great  advantage. 

Relative  Viscosity  Measurement. — On  account 
of  the  labor  involved  in  obtaining  the  dimensions 
of  the  viscometer,  many  investigators  have  followed 
the  example  of  Pf  ibram  and  Handl  in  disregarding 
these  dimensions,  and  calibrating  the  instrument 
with  some  standard  liquid.  The  most  important 
instrument  of  this  class  is  that  of  Ostwald,  Fig.  28. 
It  "consists  essentially  of  a  U-tube  with  a  capillary 
in  the  middle  of  one  limb  above  which  is  placed  a 
bulb.  A  given  volume  of  liquid  is  placed  in  the 
instrument  and  the  time  measured  that  is  required 
for  the  meniscus  to  pass  two  marks  one  above  and 
one  below  the  bulb  under  the  influence  of  the  hydrostatic 
pressure  of  the  liquid  only. 

If  770  is  the  viscosity  of  the  standard  liquid  and  77  that  of  liquid 
to  be  measured,  we  have  from  Eq.  (22) 
rj    =       Cpt  -  C'p/t 

and  if  77  is  very  nearly  equal  to  7/0  or  if  t  and  t0  are  very  large,  this 
may  be  written 

?  =  -4,  (23) 

170        Po*o 


FIG.  28. — 
The  Ostwald 
viscometer. 


76 


FLUIDITY  AND  PLASTICITY 


FIQ.  29. — Viscom- 
eter  suitable  for  the 
relative  measure- 
ment of  not  too 
viscous  liquids. 


The   pressure   in   this    instrument    must    be 
proportional  to  the  densities  so  that 


which  is  the  formula  suggested  by  Ostvvald. 
The  formula  is  true  for  dilute  solutions  when 
water  is  taken  as  the  standard,  for  77  is  then 
nearly  equal  to  rj0. 

It  is  inconvenient  to  make  the  time  of  flow 
very  large  both  on  account  of  the  lack  of 
economy  and  because  of  the  increased  danger 
of  clogging.  Unfortunately  this  formula  has 
been  used  where  neither  of  the  necessary  con- 
ditions was  complied  with  and  the  results  are 
therefore  of  uncertain  value.  It  is  much 
better  to  make  the  correction  for  the  kinetic 
energy,  in  such  cases,  than  to  attempt  to  make 
the  correction  negligible. 

It  is  a  disadvantage  of  the  Ostwald  instru- 
ment that  the  pressure  is  not  variable  at  will, 
because  if  the  time  of  flow  is  sufficient  in  one 
liquid,  in  another  more  viscous  liquid  the  time 
of  flow  may  be  intolerably  long,  practically 
necessitating  the  use  of  a  variety  of  instru- 
ments. Furthermore  the  total  pressure  is  so 
small  that  a  small  error  in  the  working  volume 
may  introduce  considerable  error  into  the 
result  and  the  density  of  the  liquid  must  be 
known  with  considerable  accuracy. 

A  form  of  instrument  which  has  the  mani- 
fest advantages  of  the  Ostwald  instrument 
and  overcomes  the  above  objections  is  shown 
in  Fig.  29.  The  volume  K  is  made  as  nearly 
as  possible  equal  in  volume,  similar  in  shape, 
and  at  the  same  height  as  C.  The  working 
volume  is  contained  between  A  and  H  and 
the  volume  of  flow  between  B  and  D,  the 
measurement  being  made  as  the  meniscus 
passes  either  from  B  to  D  or  from  D  to  B 


THE  VISCOMETER 


77 


•depending  upon  the  direction  of  the  flow.  The  corrections  are 
made  as  for  absolute  measurements  and  the  viscosity  calculated 
from  formula  (22).  In  obtaining  the  pressure  correction  due  to 
the  average  resultant  hydrostatic  pressure  in  the  viscometer  C" 
can  be  estimated  accurately  enough  by  means  of  rough  measure- 
ments. The  value  of  C  can  be 
obtained  accurately  enough  for  the 
calculation  of  this  correction  by 
assuming  p0  =  p.  After  obtaining 
the  value  of  the  hydrostatic  head  x 
in  this  way,  the  true  value  of  C  may 
be  calculated  from  an  observation 
upon  the  time  of  flow  of  any  liquid 
whose  viscosity  is  accurately  known. 
In  the  use  of  any  relative  instru- 
ment, it  is  important  that  two  stand- 
ards be  employed  so  as  to  obtain  a 
check  upon  the  method.  For  this 
purpose  a  single  liquid  may  be  used 
at  widely  different  temperatures  or 
two  or  more  liquids  may  be  used  of 
widely  different  viscosities.  While 
this  test  is  very  simple  and  its 
importance  is  obvious,  it  does  not 
appear  to  have  been  frequently 
employed. 

Viscosity  Measurements  of 
Liquids  above  the  Boiling-point.— 
If  the  viscosity  of  liquids  is  to  be 
measured  above  the  ordinary  boiling 
temperature,  one  must  work  at  pressures  above  the  atmospheric 
pressure.  The  three-way  cocks  in  Fig.  22  must  lead  to  a  low- 
pressure  reservoir,  this  pressure  being  measured  by  a  second 
manometer.  The  rubber  connections  must  of  course  be  replaced 
by  others  capable  of  withstanding  the  desired  pressure. 

Viscosity  Measurement  of  Very  Viscous  Substances. — Sub- 
stances like  pitch  which  are  excessively  viscous  can  yet  be 
measured  by  the  efflux  method  by  the  use  of  very  great  pressure 
(cf.  Barus  (1893)).  On  account  of  the  lack  of  proper  drainage, 


Section  V-Yl 


FIG.  30. — Plastometer.  For  use 
with  very  viscous  or  with  plastic 
substances. 


78 


FLUIDITY  AND  PLASTICITY 


the  apparatus  described  above  is  unsuited.  But  in  this  case  the- 
volume  may  very  properly  be  obtained  from  the  weight  of  the 
efflux  into  air,  because  the  effect  of  surface  tension  would  be 


FIG.  31. — Viscometer  for  gases  after  Schultze. 

negligible  at  these  high  pressures.  A  viscometer  designed  for 
very  viscous  substances  is  shown  in  Fig.  30.  The  use  of  this 
form  of  apparatus  is  described  in  detail  in  connection  with 
plastic  flow  (cf.  Appendix  B,  p.  320). 

The  Viscosity  Measurement  of  Gases. — A  very  satisfactory 
apparatus  for  the  measurement  of  the  viscosity  of  gases  by  the 


THE  VISCOMETER  79 

efflux  method  has  been  worked  out  through  the  labors  of  Graham 
(1846-1861),  0.  E.  Meyer  (1866-1873),  Puluj  (1876),  E.  Wiede- 
mann  (1876),  Breitenbach  (1899),  and  Schultze  (1901).  We  may 
describe  briefly  the  form  used  by  Schultze  as  illustrating  the 
modifications  which  are  necessary  in  the  apparatus  used  for 
liquids.  In  Fig.  31  the  glass  capillary,  I  =  52.54  cm,  R  = 
0.007572  cm,  is  contained  in  the  upper  chamber  of  the  bath  /, 
which  is  maintained  at  constant  temperature  by  water,  water 
vapor,  or  aniline  vapor.  A  condenser  is  shown  at  6  and  SS  is  a 
shield  to  protect  the  rest  of  the  apparatus  from  the  radiation. 
On  either  side  of  the  bath  the  apparatus  is  exactly  similar,  so  that 
only  the  right  side  is  shown  in  the  figure.  The  gas  is  contained 
in  the  bulbs  P  and  Q  (and  P'  and  Q'  on  the  left  side)  surrounded 
by  a  separate  bath.  The  lower  bulbs  are  each  connected  with 
two  stop  cocks  B  and  C  (or  B'  and  C") ;  from  B  (or  B'}  a  rubber 
tube  leads  to  the  mercury  reservoir  G  (or  G'),  and  from  C  (or  C') 
there  is  a  glass  tube  drawn  out  into  a  capillary.  Adjacent  to  both 
the  capillary  and  the  bulbs,  considerable  lengths  of  glass  tubing 
are  put  in  connection  and  immersed  in  the  respective  baths  in 
order  that  the  gas  in  the  capillary  or  bulbs  may  be  at  the  desired 
temperature  at  the  time  of  measurement.  In  each  tube  leading 
from  the  bulbs  to  the  capillary  there  is  a  stop  cock  A  (and  A') 
and  a  connection  with  a  manometer  K  (and  K'}.  By  means  of 
stop  cocks  at  E  and  E'  the  two  manometers  may  be  connected 
together  or  gas  admitted  to  the  apparatus  from  outside.  Since 
the  presence  of  water  vapor  is  objectionable  and  gases  are 
more  or  less  soluble  in  water,  the  manometer  contains  both  mer- 
cury and  water,  and  is  calibrated  before  use. 

In  makin'g  a  measurement,  enough  gas  is  admitted  into 
the  evacuated  apparatus  so  that  at  atmospheric  pressure,  the 
surface  of  the  mercury  is  in  the  lower  part  of  the  bulb  Q  and  in 
the  middle  part  of  the  bulb  Q' '.  The  stop  cock  A  is  then  closed 
and  the  mercury  reservoirs  G  and  Gf  raised,  but  the  former 
enough  higher  than  the  latter  so  that  a  pressure  head  is  estab- 
lished which  is  a  few  millimeters  greater  than  is  desired  in  the 
measurement.  The  mercury  fills  the  two  bulbs  Q  and  Q'.  When 
the  temperature  is  constant  the  stop  cock  A  is  opened.  The 
pressure  is  immediately  adjusted  and  thereafter  maintained 
constant  by  means  of  the  screws  F  and  F1  which  serve  to  slowly 


80  FLUIDITY  AND  PLASTICITY 

raise  or  lower  the  mercury  reservoirs.  When  the  mercury  passes 
into  the  bulb  P,  contact  is  formed  with  a  platinum  point  and  an 
electrical  signal  given.  At  this  moment  the  chronometer  is 
started.  After  the  elapse  of  sufficient  time,  the  stop  cock  B  is 
closed  and  thus  the  current  is  broken  between  the  two  platinum 
electrodes  at  either  side  of  this  stop  cock,  and  a  signal  is  given. 
The  mercury  is  now  allowed  to  run  out  through  the  stop  cock  C 
until  the  signal  is  given  when  the  mercury  loses  connection  with 
the  platinum  point  in  the  bulb  P.  From  the  weight  of  this 
mercury,  the  volume  of  flow  is  calculated. 


PART  II 

FLUIDITY  AND  OTHER  PHYSICAL  AND 
CHEMICAL  PROPERTIES 

CHAPTER  I 
VISCOSITY  AND  FLUIDITY 

It  has  been  tacitly  assumed  by  the  great  majority  of  workers 
that  when  two  liquids  are  mixed,  the  viscosity  of  the  mixture 
is  normally  a  linear  function  of  the  composition.  This  appeared 
as  early  as  1876  in  the  work  of  Wijkander.  In  a  great  many 
mixtures,  including  practically  all  of  those  in  which  water  is  a 
component,  the  viscosity  is  certainly  very  far  from  being  a  linear 
function  of  the  composition,  there  being  often  a  maximum  in  the 
viscosity  curves.  However  water  mixtures  should  not  be  con- 
sidered as  "normal,"  but  since  it  is  difficult  to  decide  what  shall 
be  considered  normal  mixtures,  the  question  whether  the 
viscosities  are  additive  or  not  is  admittedly  difficult  of  solution. 
Dunstan  (1905)  classifies  as  normal  those  mixtures  whose  vis- 
cosity-weight concentration  curves  do  not  show  a  maximum  or 
a  minimum.  This  classification  is  not  satisfactory  not  only 
because  it  lacks  a  theoretical  justification  but  also  because  many 
of  the  so-defined  normal  mixtures  give  curves  which  depart 
considerably  from  the  linear,  so  that  the  suspicion  is  aroused  that 
the  occurrence  of  a  maximum  or  minimum  may  depend  upon 
accidental  circumstances  such  as  the  nearness  to  equality  of  the 
viscosity  of  the  components.  The  accidental  character  of  such 
a  classification  is  very  striking  in  mixtures  which  fall  into  the 
normal  class  at  one  temperature  but  at  a  slightly  different  tem- 
perature must  be  classified  as  abnormal. 

Such  light  as  can  be  gained  from  a  study  of  the  viscosities 
of  mixtures,  seems  to  lead  to  the  conclusion  that  viscosities  are! 
not   additive,    as   has   been    assumed.     Thus    Dunstan    (1904) 
remarks,  "The  law  of  mixtures  is  never  accurately  obeyed  and 

6  81 


82  FLUIDITY  AND  PLASTICITY 

divergences  from  it  seem  to  be  more  clearly  marked  out  in  the 
case  of  viscosity  than  with  other  properties,  such  as  refractive 
index."  Thorpe  and  Rodger  (1897)  say,  "The  observations 
described  in  this  paper  afford  additional  evidence  of  the  fact  indi- 
cated by  Wijkander  and  supported  by  Linebarger,  that  the  vis- 
cosity of  a  mixture  of  miscible  and  chemically  indifferent  liquids 
is  rarely,  if  ever,  under  all  conditions,  a  linear  function  of  the 
composition.  It  seldom  happens  that  the  liquid  in  a  mixture 
preserves  the  particular  viscosity  it  posesses  in  the  unmixed 
condition.  To  judge  from  the  instances  heretofore  studied,  the 
viscosity  of  the  mixture  is,  as  a  rule,  uniformly  lower  than  the 
mixture  law  would  indicate,  but  no  simple  relation  can  yet  be 
traced  between  the  viscosity  of  a  mixture  and  that  of  its  constit- 
uents." Thorpe  and  Rodger  were  so  struck  by  the  absence  of 
linearity  in  the  viscosity  curves,  that  they  thought  that  an  ex- 
planation was  needed  for  the  fact  that  the  viscosity  curves  of 
some  mixtures  measured  by  Linebarger  (1896)  are  indeed  linear. 
"  The  observed  viscosities  in  general  are  less  than  those  calculated 
by  the  mixture  rule,  except,  possibly,  in  the  case  of  mixtures  of 
benzene  and  chloroform  and  mixtures  of  carbon  disulfide  with 
benzene,  toluene,  ether,  and  acetic  ether,  where,  possibly,  the 
temperature  of  observation  (25°)  was  too  near  the  boiling-point 
of  the  carbon  disulfide  to  make  any  specific  influence,  which  that 
liquid  might  exert  at  lower  temperatures,  perceptible." 

Lees  (1900)  showed  what  are  the  necessary  assumptions  in 
regard  to  the  nature  of  flow  in  mixtures,  so  that  the  viscosities 
should  be  additive,  but  by  making  a  careful  study  of  existing 
data,  he  found  little  justification  for  these  assumptions.  Simi- 
larly Lees  tried  the  assumptions  that  fluidities  or  logarithmic 
viscosities  are  the  characteristic  additive  property,  but  he  was 
unable  to  obtain  a  satisfactory  verification  of  either  from  the 
experimental  results. 

The  question  before  us  seems  to  be:  "Is  viscosity  or  fluidity 
or  some  function  of  one  of  them  the  characteristic  additive  prop- 
erty?" The  answer  to  this  question  is  imperative  before  we 
can  intelligently  discuss  the  relation  of  viscosity  to  other  proper- 
ties. This  statement  requires  no  proof  in  view  of  the  statements 
which  we  have  quoted  to  show  that  in  some  cases  the  viscosity 
concentration  curve  is  linear  according  to  assumption,  but  in  the 


VISCOSITY  AND  FLUIDITY  83 

great  majority  of  cases  it  is  sagged  and  there  is  110  known  law  to 
account  for  the  peculiarity.  Surely  any  discussion  of  chemical 
combination  or  of  dissociation  on  the  basis  of  deviation  from  the 
"  normal"  curve  under  such  conditions  would  be  of  very  uncertain 
value. 

There  are  numerous  reciprocal  relations  besides  viscosity  and  i 
fluidity,  such  as  electrical  resistance  and  conductance,  or  specific/ 
heat  and  heat  capacity,  or  specific  gravity  and  specific  volume.  \ 
It  has  been  repeatedly  pointed  out1  that  if  one  of  these  is  additive,  / 
its  reciprocal  cannot  be.  It  is  singular  enough  that  among  all 
of  these  reciprocal  relations,  viscosity  is  the  only  one  for  which 
the  decision  has  not  been  reached  as  to  whether  viscosity  is 
additive  or  not,  or  if  it  is,  under  what  conditions.  In  electricity 
for  example  we  have  absolutely  no  doubt  but  that  resistances 
are  additive  under  certain  conditions,  viz.,  when  the  conductors 
are  in  series,  and  likewise  that  conductances  are  additive  under 
other  equally  definite  conditions,  viz.,  when  the  conductors 
are  in  parallel.  It  seems  probable  that  the  present  unsatisfactory 
condition  as  regards  viscosity  has  arisen  due  to  the  extraordinary 
sensitiveness  of  this  property  to  molecular  changes  in  fluids, 
either  combination  or  dissociation.  We  shall  attempt  to  reach 
a  solution  of  the  problem  from  a  consideration  of  the  nature  of 
viscous  flow  and  then  test  this  solution  by  means  of  the  experi- 
mental facts.  After  we  have  reached  a  conclusion  in  regard  to  the 
true  additive  property  under  given  conditions,  it  may  well  turn 
out  that  the  present  unsatisfactory  condition  will  prove  to  be  a 
blessing  in  disguise,  for  it  may  then  be  shown  that  viscosity  is  of 
the  greatest  importance  in  physiochemical  investigations. 

The  fundamental  law  of  viscous  flow 
dv  =F 
dr        rj 

is  the  analogue  of  the  well-known  electrical  law  of  Ohm.  In 
fact  Elie  in  1882  suggested  a  modification  of  the  Wheatstone's 
bridge  method  for  the  measurement  of  viscous  resistance. 

Case  I.  Viscosities  Additive — Emulsions. — We  will  first  con- 
sider the  very  simple  case  of  a  series  of  vertical  lamellae  of  viscous 
material  arranged  alternately,  as  in  Fig.  32,  and  subjected  to  a 

1  Cf.  p.  89. 


84  FLUIDITY  AND  PLASTICITY 

horizontal  shearing  stress.  For  convenience  suppose  that  all 
of  the  lamellae  of  the  one  substance  A  have  the  same  thickness 
si  and  that  the  laminae  of  the  substance  B  have  the  uniform 
thickness  82,  etc.  Let  the  viscosities  of  the  substances  be  -n\, 
772  ...  and  the  shearing  stresses  per  unit  area  pi,  p%  .  .  . 
respectively;  then  if  R  is  the  distance  between  the  horizontal 
planes,  the  velocity  of  the  moving  surface  is 
_  RP  _  Rpi  _  Rpz 

V    —      TT     —  — 

a          r?i          172' 

where  H  is  the  viscosity  of  the  mixture,  and  P  is  the  average 
shearing  stress  over  the  entire  distance  S. 


'1 

| 

?y                J^r\A.V 

"J 

/            hence 

/ 

/'// 

• 

-XX 

'  & 

V 

. 

r                                 TT    ^  //^l"«-       i^   /^2"2      T      •       •       • 

"  ~  o  \              5  ~ 

Fig.  32.—  Diagram  to  illus- 

trate  additive  viscosities.    But  since  Si/o  is  the  fraction  by  volume 
of  the  substance  A  present  in  the  mixture, 
which  we  may  designate  a,  and  similarly  s2/S  =  6,  etc., 

H  =  a-ni  +  &T?2  +  .    .    .  (24) 

This  case  is  of  particular  interest  in  connection  with  emulsions 
and  many  other  poorly  mixed  substances.  The  formula  tells 
us  that  the  viscosity  of  the  mixture  is  the  sum  of  the  partial 
viscosities  of  the  components,  provided  that  the  drops  of  the 
emulsion  completely  fill  the  capillary  space  through  which  the 
flow  is  taking  place. 

Case  II.  Fluidities  Additive  —  Fluid  Mixtures.  —  If  the  lamellae 
are  arranged  parallel  to  the  direction  of  shear,  as  shown  in  Fig. 
33,  we  have  a  constant  shearing  stress,  so  that 

p=W=M_2=    .  (24a) 

TI         r2 

where  vi,  vz,  .  .  .  are  the  partial  velocities  as  indicated  in  the 
figure. 

There  are  two  different  ways  of  defining  the  viscosity  of  a 
mixture,  and  it  becomes  necessary  for  us  to  adopt  one  of  these 
before  we  proceed  further. 

1.  If  we  measure  viscosity  with  a  viscometer  of  the  Coulomb 


VISCOSITY  AND  FLUIDITY 


85 


or  disk  type,  we  actually  measure  the  velocity  v,  BS  in  the  figure, 
and  we  very  naturally  assume  that 


P  = 


(24b) 


2.  It  is  more  usual,  however,  to  calculate  the  viscosity  from 
the  volume  of  flow,  as  in  the  Poiseuille  type  of  instrument. 
Let  vr,  BS'  in  the  figure,  be  the  effective  velocity  which  the 
surface  BS  would  have,  were  the  series  of  lamellae  replaced  by 


S'  S    S" 


FIG.  33. — Diagram  to  illustrate  additive  fluidities. 


a  homogeneous  fluid  having  the  same  volume  of  flow.  The 
effective  velocity  is  related  to  the  quantity  of  fluid  U  passing  per 
second  in  a  stream  of  unit  width,  as  follows: 

U  =    V'^ 

Let  the  viscosity  as  calculated  from  the  flow,  as  for  a  homo- 
geneous fluid,  be  H',  then 

P  =    »V-  =  W"  (24c) 

It  is  to  be  noted  that  had  the  less  viscous  substance  been  in 
contact  with  the  surface  AE,  the  effective  velocity  of  flow  would 
have  been  represented  by  the  distance  BS".  We  shall  take  the 
former  of  these  for  our  definition  of  the  viscosity  of  a  mixture, 


86  FLUIDITY  AND  PLASTICITY 

since,  as  we  shall  now  show,  by  using  it  the  viscosity  is  indepen- 
dent of  the  number  or  arrangement  of  the  lamellae. 

Since  v  =  v\  +  v2  +  .    .    . 
we  obtain  from  Eqs.  (24a)  and  (24b)  that 


etc. 
/i  K 

the  fluidity  of  the  mixture  is 


.    .  (25) 

The  fluidities  are,  according  to  this  definition,  strictly  additive 
and  entirely  independent  of  the  number  and  arrangement  of 
the  layers.  Since,  however,  the  viscosities  are  usually  calculated 
by  means  of  the  Poiseuille  formula  based  on  the  volume  of  flow, 
it  is  important  to  determine  for  a  given  arrangement  of  lamellse 
what  correction  must  be  made  to  the  effective  viscosity,  as  calcu- 
lated from  the  volume  of  flow,  to  make  it  accord  with  the  true 
viscosity,  as  defined  above  and  as  obtained  by  the  disk  or  other 
similar  method  for  the  measurement  of  viscosity. 
Reverting  again  to  the  figure,  we  find  that 


+  Vir2  +  v2r2  +  ViTz  +  -~ 

If  there  were  n  pairs  of  alternate  lamellae  of  the  two  substances 
A  and  B 

U  =    [n%iri  +  n(n  +  I>ir2  +  nv\r2  +  n(n  -  l.W,]  (26). 


D 

Since  n  =  -      —  ,  on  substituting  into  Eq.  (26)  the  values  of 
fi  T  TI 

Vi  and  v2,  we  get 

TT    R2pr  .  a&/ 

U  ••    —y-^api  +  6^2  +  ~\Vi  — 

and  if  «S>/  =  Trf  we  obtain  from  Eq.  (24b) 
ti 

V  =  an  +  6^2  +      (Vi  -  vj  (27) 


VISCOSITY  AND  FLUIDITY 
and  when  n  =  °°,  the  fluidity  becomes  simply 


87 


and  in  this  case 

*'  -  *.  (28) 

In  a  homogeneous  mixture  it  appears,  therefore,  that  the  two 
definitions  lead  to  the  same  fluidity,  and  experimental  results 
lead  us  to  believe  that  this  is  the  case  usually  presented  in  liquid 
mixtures,  since  the  disk  method  and  the  capillary  tube  method 

give  the  same  fluidity  so  far  as  we  > 

have  certain  knowledge.  If,  however, 
the  number  of  lamella  is  small,  which 
may  well  be  the  case  in  very  imperfect 
mixtures,  or  when  the  flow  takes  place 
through  very  narrow  passages,  the 
effective  fluidity  as  calculated  from 
the  volume  of  flow  may  be  either 
greater  or  less  than  the  sum  of  the 
partial  fluidities  of  the  components, 


FG- 


necessarily  brings  about  com- 

piete  mixing,  so  that  even  when 

the  viscosities  were  originally 


depending    upon    the    order    of    the    trate 

arrangement  of  the  lamella  in  refer-    mixed  but  miscibie  fluids,  flow 

,i  ,  .  c  mi. 

ence  to  the  stationary  surface.     The 

amount    to    be    added    or    subtracted 

from  the  effective  fluidity  in  order  to 
obtain  the  true  fluidity  is  represented  cible  fluids,  the  layers  A  and 
by  the  term,  corresponding  to  the  L'SSJSSS'l.'SST 
areas  A  CD,  etc.  or  AFD,  etc.,  Fig.  33. 

A  combination  of  the  cases  I  and  II  would  lead  to  a  checker- 
board arrangement,  but  it  may  be  shown  now  that  such  an 
arrangement  tends  to  reduce  itself  to  the  case  II  where  fluidities 
are  additive. 

If  the  arrangement  considered  in  Fig.  32  is  subjected  to 
continued  shearing  stress,  the  lamellae  will  tend  to  become 
indefinitely  elongated  as  indicated  in  Fig.  34;  and  unless  the 
surface  tension  intervenes,  as  may  be  the  case  in  immiscible 
liquids,  the  lamellae  will  approach  more  and  more  nearly  the 
horizontal  position.  Thus,  so  far  as  we  can  determine  without 
going  into  the  complicated  problem  of  the  molecular  motions, 
it  seems  certain  that  the  fluidities  will  become  more  and  more 


88  FLUIDITY  AND  PLASTICITY 

nearly  additive  as  the  flow  progresses  and  the  mixture  becomes 
more  and  more  nearly  complete.  This  result  takes  place  further- 
more irrespective  of  the  original  arrangement  of  the  parts  of  the 
mixture. 

Some  one  may  object  that  a  perfectly  homogeneous  mixture 
— in  itself  a  contradiction  of  terms — is  not  made  up  of  layers 
such  as  we  have  considered  in  these  greatly  simplified  cases. 
There  can  be  no  doubt  whatever  of  the  existence  of  layers  during 
the  process  of  mixing.  No  one  has  watched  the  drifting  of 
tobacco  smoke  in  his  study  without  noting  how  it  is  drawn  out 
into  gossamer-like  layers.  Since  the  fluidity  is  greatest  when 
fluidities  are  additive,  there  would  have  to  be  a  sudden  drop  in 
fluidity  as  the  mixture  became  perfect,  if  the  fluidities  were  no 
longer  additive.  This  is  not  supported  by  any  experimental 
evidence. 

We  have  already  noted  that  when  there  is  no  chemical  action 
between  the  components  of  a  mixture,  the  viscosity-concentration 
curves  are  usually  but  not  always  sagged.  Dunstan  (1913)  has 
put  it:  "It  can  therefore  safely  be  predicted  that  wherever  the 
two  components  show  little  tendency  for  chemical  union  a 
sagged  curve,  or  one  departing  but  slightly  from  linearity,  will 
be  found."  If  the  fluidities  of  such  mixtures  are  additive,  these 
facts  ought  to  be  accounted  for  by  the  theory,  peculiar  as  they 
may  seem  to  be.  We  shall  first  prove  that  according  to  the 
theory  that  fluidities  are  additive,  we  should  expect  the  viscosity- 
concentration  curves  to  be  sagged. 

Equations  (25)  and  (24)  represent  the  two  assumptions 
that  fluidities  are  additive  and  that  viscosities  are  additive 
respectively  but  for  convenience  we  shall  assume  that  only  two 
components  are  present  in  the  mixture.  From  Eq.  (24)  we  get 
that 


b<p\  +  a<f>2 

When  a  =  0  or  1,  and  6  =  1  or  0  respectively,  <p  must  be  equal  to 
a>'.     For  all  intermediate  values  of  a  and  b  we  desire  to  learn 


VISCOSITY  AND  FLUIDITY  89 

whether  <p  must  be  invariably  greater  than,  equal  to,  or  less 
then  p'.     Multiplying  Eq.  (25)  by  unity,  we  obtain 


Since  6  =  1  —  a, 

2a(a  -  1W2  +  a(l  -  a)  (<pS  +  ^22)|o 
Discarding  the  known  roots,  a  =  0  and  a  =  1,  we  get 

<f>l2    —    2<pl<?2    +   <£>22    -     0, 

which  is  a  perfect  square  and  therefore  must  be  positive.  Hence, 
when  <pi  is  equal  to  <pz,  <p  is  equal  to  <?',  but  for  all  other  values 
<p  must  be  greater  than  ^>/.  Our  conclusions  may  be  stated  as 
follows  : 

1.  The  viscosity  of  a  thorough  mixture  of  chemically  indifferent 
fluids  must  always  be  less  than  would  be  expected  on  the  assump- 
tion that  viscosities  are  additive,  but  this  inequality  will  approach 
zero  as  the  difference  between  the  viscosities  of  the  components 
approaches  zero. 

2.  So,  on  the  other  hand,  the  viscosity  of  an  emulsion  must  be 
greater  than  that  of  a  perfect  mixture  of  the  same  composition, 
because  in  emulsions  the  viscosities  tend  to  become  additive. 

Equation  (25)  may  be  expressed  in  the  form 

9  =  <f>i  +(<P2  —  <f>i)b,  (29) 

where  <pi  and  <pz  are  constant  and  <p  and  6  are  variable.  The 
corresponding  viscosity  equation  is 

•n  =  —^-~r  -  ^  (3°) 

<f>i  +  (<P*  —  <pi)o 

It  is  important  to  note  that  Eq.  (29)  is  the  equation  of  a 
straight  line,  but  that  Eq.  (30)  is  the  equation  of  a  hyperbola. 
If  we  replace  <pi  +(<p2  —  <f>i)b  by  (<pa  —  <pi)m,  where 


we  get 

1 

mi]  =  - 

<f>2    —    <p\ 

which  is  the  equation  of  an  equilateral  hyperbola,  whose  X-axis  is 


90  FLUIDITY  AND  PLASTICITY 

at  a  distance  <p\/(<p*  —  <p\)  to  the  left  of  the  origin  to  which 
Eq.  (27)  is  referred. 

3.  We  conclude  therefore  that  the  curve  obtained  by  plotting 
viscosities  against  volume  concentrations  is  not  normally  linear 
but  a  part  of  an  equilateral  hyperbola. 

From  Eq.  (30)  we  find  the  curvature  for  any  mixture  to  be 


By  differentiating  this  curvature  in  respect  to  the  concentration 
a  and  equating  to  zero,  we  find  the  concentration  where  the 
curvature  is  a  maximum  to  be 


a 


Substituting  this  value  in  Eq.  (31),  the  amount  of  the  curvature 
where  the  curvature  is  a  maximum,  is  found  to  be 


(33) 

4.  The  curvature  of  the  viscosity-volume  concentration  curves 
s  greatest  when  the  difference  between  the  fluidities,  i.e.,  <p2  —  <pi, 
s  large,  and  becomes  zero  when  <p2  —  <pi  =  0. 

5.  The  curvature  must  continually  decrease  as  the  concentra- 
tion increases  unless  the  square  root  of  <p2  —  <PI  is  greater  than  <pi, 
in  which  case  the  point  of  greatest  curvature  will  be  found  at 
some  positive  concentration  (cf.  Eq.  (32)). 

6.  Mathematically  considered,  the  curvature  is  dependent  only 
upon  the  difference  in  the  fluidities  of  the  components,  i.e., 
<?2  —  <pi  and  not  upon  <pi,  but  since  we  can  only  realize  positive 
values  of  a,  it  follows  that  for  a  given  value  of  <p2  —  <?i  the 
curvature  at  any  concentration  will  be  greatest  when  <p\  is  very 
small. 

LIQUID  MIXTURES 

The  first  conclusion  is  confirmed  by  the  repeated  observation 
that  viscosity-concentration  curves  of  homogeneous  mixtures  are 
normally  sagged.  The  fourth  conclusion  offers  an  explanation 
of  the  fact  that  they  are  sometimes  very  nearly  linear.  In 
particular,  this  conclusion  is  confirmed  by  Thorpe  and  Rodger, 
who,  in  commenting  on  the  data  of  Linebarger,  make  the  interest- 


VISCOSITY  AND  FLUIDITY 


91 


ing  observation  "As  a  rule,  the  greater  the  difference  between  the 
viscosities  of  the  pure  liquids,  the  greater  is  the  difference  between 
the  calculated  and  the  observed  values  of  the  mixtures."  The 
second  conclusion  is  confirmed  by  the  sudden  drop  in  the  fluidity 
of  a  mixture  as  it  is  cooled  below  its  critical-solution  tempera- 
ture. This  has  often  been  noted  and  commented  upon,  and  will 
be  discussed  more  fully  at  a  later  point.  In  undercooled  liquids 


450 
400 
350 
.300 


wo 

200 
SO 


ACTS 


.0025 


60 


80 


100 


Weight  percentage  of  second  component  of  mixture. 

Fia.  35. — 1.  Fluidity  curve  of  nitrobenzene  and  ethyl  acetate  at  25°;  2. 
Fluidity  curve  of  ethyl  alcohol  and  acetone  at  25°;  3.  Fluidity  curve  of  benzene 
and  ethyl  acetate  at  25°;  4.  Fluidity  curve  of  benzene  and  ethyl  ether  at  25°; 
5.  Fluidity  curve  of  carbon  bisulphide  and  ethyl  ether  at  25°;  2rj.  Viscosity  curve 
(dotted)  of  ethyl  alcohol  and  acetone  at  25°.  Were  viscosities  additive,  this 
curve  would  be  linear  (dashes). 

and  other  very  viscous  substances  it  has  been  often  noted  that  the 
viscosity  curves  have  a  very  high  degree  of  curvature,  at  least 
during  a  part  of  their  course.  This  is  in  harmony  with  the  fifth 
and  sixth  conclusions. 

Were  the  fluidity-volume  concentration  curves  invariably 
linear,  it  would  constitute  an  experimental  verification  of  our 
fourth  conclusion.  Unfortunately  for  this  purpose,  the  fluidity 
concentration  curves  are  rarely,  if  ever,  perfectly  linear,  for  the 
reason  that  has  been  indicated;  viz.  that  there  is  perhaps  nearly 
always  some  molecular  change  on  mixing,  even  though  very 
feeble,  and  to  this  change  fluidity  is  very  sensitive.  These 


92  FLUIDITY  AND  PLASTICITY 

changes  are  of  no  interest  to  us  at  this  point,  but  the  fact  is  very 
important  for  us  that  the  fluidity-volume  concentration  curves 
are  much  more  nearly  linear  than  the  viscosity  concentration 
curves.  To  become  convinced  of  this  the  reader  should  plot 
the  fluidity  concentration  curves  of  several  of  the  mixtures  given 
by  Wijkander  (1878),  Linebarger  (1896),  Dunstan  et  cet.  (1904) 
and  others  and  compare  them  with  the  viscosity  curves  given 
by  those  authors.  A  few  of  these  curves  are  given  in  Fig.  35.  * 
Kendall  (1913)  has  gone  over  the  whole  range  of  available  data 
and  finds  that  the  percentage  deviation  between  the  observed 
and  calculated  values  is  11.1  for  the  viscosity  curves  and  3.4  for 
the  fluidity  curves. 

FLUIDITY  AND  TEMPERATURE 

The  conclusion  that  fluidities  are  additive  has  far-reaching 
consequences,  so  that  there  arise  tests  for  the  conclusion  which 
were  at  first  quite  unsuspected.  For  example,  it  is  evident  that 
the  reasoning  which  has  been  found  to  hold  for  mixtures  of  fluids 
must  also  hold  for  mixtures  of  the  same  fluid  at  different  tempera- 
tures; for  a  fluid  at  any  temperature  may  be  thought  of  as  a 
mixture  of  appropriate  amounts  of  portions  of  the  fluid  main- 
tained at  the  extreme  temperatures.  Hence,  we  are  led  to  the 
hypothesis  that  the  fluidity-temperature  curves  of  pure  fluids 
should  normally  be  linear,  and  the  viscosity-temperature  curves 
hyperbolic.  This  relation  cannot  hold  through  a  change  of  state 
because  a  new  cause  of  viscosity  then  enters  in.  Furthermore  the 
fluidity  of  liquids  is  closely  related  to  their  volumes,  as  we  shall 
see  later,  and  the  volumes  of  liquids  do  not  generally  increase  in 
a  linear  manner  with  the  temperature.  Then,  too,  association  and 
dissociation  may  play  a  disturbing  factor,  so  that  as  in  mixtures, 
a  perfect  verification  can  scarcely  be  expected.  In  Fig.  36  there 
are  given  the  fluidity-temperature  and  viscosity-temperature 
curves  of  mercury  and  of  water  from  0  to  100°C.  Both  of  the 

1  We  have  here  but  a  rough  test  of  the  truth  of  the  hypothesis  that 
fluidites  are  additive  in  homogeneous  mixtures,  because  the  fluidites  of  the 
components  are  too  close  together,  all  of  the  components  are  certainly  not 
inert,  and  volume  concentrations  should  have  been  studied.  More  rigorous 
tests  of  the  hypothesis  will  be  made  after  the  law  of  Batschinski  has  been 
considered. 


VISCOSITY  AND  FLUIDITY 


93 


fluidity  curves  are  much  more  nearly  linear  than  are  the  viscosity 
curves,  the  true  linear  curve  being  represented  in  each  case  by  a 
series  of  dashes.  Mercury  is  an  ideal  substance  in  this  connec- 
tion for  it  is  far  removed  from  the  critical  temperature,  it  is  not 
highly  associated,  and  its  volume  increases  in  a  linear  manner 
with  the  temperature.1  The  fluidity  curve  is  almost  perfectly 
linear,  what  curvature  there  is  being  in  a  direction  opposite  to 
that  of  every  other  known  substance,  so  that  it  can  hardly  be 


cr      20"       4tr      to"       iff       ioo' 

Temperature  Centigrade. 

FIG.  36.  —  Fluidity   (continuous)   and  viscosity  (dotted)  temperature  curves  for 
mercury  and  water. 

regarded  as  certain  that  this  deviation  is  not  due  to  experimental 
error.  An  extensive  study  of  the  fluidity-temperature  curves  of 
pure  liquids  leads  to  the  conclusion  that  even  when  the  expansion 
is  not  linear  and  there  is  association,  the  curves  approach  linearity, 
as  is  seen  to  be  the  case  with  water  in  the  figure.  The  extent  to 
which  this  is  true  can  be  best  judged  by  an  algebraic  analysis 
of  the  data  to  be  given  later.  However  it  may  be  stated  here 
that  the  first  approximation  of  Meyer  and  Rosencranz  (1877) 

'  -  (34) 


Landolt  and  Bornstein,  Tabellen,  3d.  ed.,  p.  41. 


94  FLUIDITY  AND  PLASTICITY 

when  put  in  the  form 

V  =  A  +  BT  (35) 

is  but  an  algebraic  expression  of  the  law  that  the  fluidity  of  a 
liquid  is  a  linear  function  of  the  temperature.  The  law  is  only 
approximately  true,  but  even  with  the  alcohols  where  the  curva- 
ture is  greatest,  there  is  an  approach  to  linearity  at  high  tempera- 
tures. Like  the  Law  of  Boyle,  we  may  assume  that  this  law  holds 
in  ideal  cases,  and  that  the  theory  underlying  it  is  valid. 

EMULSIONS 

The  study  of  the  viscosity  of  mixtures  near  their  critical- 
solution  temperatures  affords  another  very  sharp  and  distinct 
means  for  testing  the  theory  which  has  been  outlined.  It  has 
been  pointed  out  that  the  fluidities  should  be  additive  in  the  per- 
fect mixture  but  the  viscosities  additive  in  the  emulsion. 
According  to  the  second  conclusion  page,  89,  there  should  be  a 
sudden  drop  in  the  fluidity  at  or  near  the  critical  solution  tempera- 
ture. We  do  not  propose  to  discuss  in  detail  here  the  viscosity 
of  colloids  but  it  is  appropriate  here  to  seek  an  answer  to  the  ques- 
tion "Has  such  a  drop  in  fluidity  ever  been  observed?" 

Ostwald  and  Stebutt  (1897)  observed  an  abnormally  large  vis- 
cosity in  a  mixture  of  isobutyric  acid  and  water  in  the  neighbor- 
hood of  the  critical-solution  temperature.  This  was  attributed 
by  them — not  to  the  reason  given  above — but  to  the  fact  that  at 
the  critical-solution  temperature,  the  surf  ace  energy  becomes  zero. 

Friedlander  (1901)  investigated  the  phenomena  which  are 
peculiar  to  the  critical-solution  temperature  in  an  intensive 
manner.  He  found  a  very  marked  increase  in  the  viscosity  as 
the  solution  was  cooled  to  temperatures  where  the  opalescence 
became  evident  and  the  critical-solution  temperature  was 
approached.  He  observed  the  opalescence  with  particular  care. 
His  investigation  was  extended  to  include  phenol  and  water, 
and  the  ternary  mixture  of  benzene,  acetic  acid,  and  water. 
Similar  relations  were  found  in  all  proving  that  the  phenomena 
are  quite  general.  He  concluded  that  the  temperature  coefficient 
of  viscosity  is  greatest  where  the  opalescence  and  the  tendency 
to  foam  are  greatest.  He  says,1  "Der  Triibungsgrad  und  Tem- 
peraturkoefficient  der  inneren  Reibung  zeigen  eine  starke  Zu- 

i  Friedlandler,  439. 


VISCOSITY  AND  FLUIDITY  95 

nahme  im  kritischen  Gebiete  und  stehen  mit  einander  in  einem 
innigen  Zusammenhange."  Friedlander  also  observed  that  the 
expansion  coefficient  and  the  coefficient  of  electrical  conductivity 
as  well  as  the  refractive  index  remained  normal.  He  believed 
that  it  was  necessary  to  go  farther  than  had  Ostwald  and  Stebutt 
in  order  to  reach  an  explanation,  and  that  a  definite  radius  of 
curvature  of  the  separating  surfaces  must  correspond  to  each 
temperature,  otherwise  the  degree  of  opalescence  could  not  be 
definitely  determined.  He  therefore  attributed  the  increase  in 
viscosity  to  the  formation  of  drops,  but  he  was  puzzled  by  the 
fact  that  when  a  solution  of  colophonium  in  alcohol  is  poured  into 
a  large  quantity  of  water,  a  highly  opalescent  liquid  is  obtained 
which  has,  nevertheless,  practically  the  same  viscosity  as  pure 
water.  This  theory  of  Friedlander  is  apparently  an  outgrowth 
of  the  theory  of  "  halbbegrenzte  Tropfen"  of  Lehmann.  Fried- 
lander  also  discussed  the  electrical  theory  of  Hardy  that  an 
increase  of  work  would  be  required  to  move  the  particles  of  a  liquid 
among  charged  particles,  so  that  if  the  "drops"  were  charged 
an  increase  in  viscosity  might  result.  But  by  experiment  Fried- 
lander  found  that  an  electrical  field  was  without  noticeable 
effect  upon  an  opalescent  liquid. 

Friedlander's  values  are  expressed  in  relative  units.  Scarpa 
(1903)  and  (1904)  has  measured  the  viscosity  of  solutions  of 
phenol  and  water,  expressing  his  results  in  absolute  units.  For  a 
given  temperature,  he  plotted  the  viscosities  against  the  varying 
concentrations,  and  obtained  a  point  of  inflection  in  the  curves 
at  the  critical-solution  temperature.  He  tried  to  explain  the 
irregularities  on  the  assumption  that  hydrates  are  formed.  He 
was  apparently  unfamiliar  with  the  work  of  Friedlander. 

Rothmund  (1908)  started  from  Friedlander's  work  to  make  a 
study  of  the  opalescence  at  the  critical  temperature.  He  meas- 
ured the  times  of  flow  of  butyric  and  isobutyric  acid  solutions  in 
water,  noting  particularly  the  effect  upon  the  opalescence  of 
adding  various  substances,  both  electrolytes  and  non-electrolytes. 
He  objected  to  the  hypothesis  of  Friedlander  in  that,  according 
to  the  well-known  formula  of  Lord  Kelvin,  small  drops  are  less 
stable  than  large  ones,  so  that  the  former  must  tend  to  disappear. 
Furthermore  he  remarked  upon  the  entirely  analogous  opales- 
cence which  is  observed  in  a  single  pure  substance  at  its  ordinary 


96  FLUIDITY  AND  PLASTICITY 

critical  temperature.  Rothmund  therefore  called  to  his  aid 
Donnan's  hypothesis  that  when  drops  are  very  small  their  surface 
tension  is  very  different  from  that  of  the  liquid  in  bulk  and  is  a 
function  of  the  radius  of  curvature.  Since  at  the  critical  tem- 
perature the  surface  tension  is  normally  zero,  it  was  thought  that 
the  small  drops  might  thus  exist  in  a  state  of  stable  equilibrium 
in  the  neighborhood  of  the  critical-solution  temperature.  As  the 
temperature  is  raised  the  opalescence  would  become  less  and  less, 
due  to  the  solution  of  the  drops.  Rothmund  found  that  the 
addition  of  naphthalene  to  his  solutions  greatly  increased  the 
opalescence,  while  the  addition  of  grape  sugar  decreased  it  very 
greatly,  although  the  effect  of  these  additions  upon  the  viscosity 
was  negligible.  He  reasoned  that  the  refractive  index  of  butyric 
acid  is  greater  than  that  of  water  and  sugar  and  electrolytes 
raise  the  refractive  index  of  water,  hence  they  make  the  presence 
of  small  drops  less  evident.  Naphthalene  does  not  dissolve  in 
water  but  does  dissolve  in  butyric  acid,  raising  its  refractive 
index,  and  therefore  it  makes  the  opalescence  more  apparent. 

Von  Smoluchowski  (1908)  regards  Rothmund's  hypothesis  as 
superfluous,  believing  that  the  kinetic  theory  is  sufficient  to 
explain  the  opalescence.  According  to  him,  differences  in 
molecular  motion,  local  differences  in  density,  and  therefore 
differences  in  surface  tension  cause  the  critical  temperature  to  be 
not  entirely  definite.  Due  to  this  indefiniteness  in  the  critical 
temperature,  rough  surfaces  are  formed,  which  must  have  a 
thickness  of  less  than  a  wave  length  of  light,  since  greater  thick- 
nesses would  not  reflect  the  light.  The  inequalities  in  the  density 
would  reach  their  maximum  at  the  critical  temperature. 

Bose  and  his  co-workers  (1907-1909)  have  also  verified 
these  earlier  observations  that  abnormally  large  viscosities  are 
obtained  at  the  critical-solution  temperature.  Bose  regards  this 
as  due  to  the  rolling  of  drops  of  liquid  along  the  capillary.  They 
did  a  considerable  amount  of  work  to  prove  that  "crystallin"  or 
"anistropic"  liquids  are  similar  to  the  emulsions  here  discussed. 
Bose  proved  that  these  liquids  have  abnormal  viscosities  near  the 
clarifying  point  and  they  also  possess  marked  opalescence. 
Vorlander  and  Gahren  had  found  that  a  crystallin  liquid  may 
result  from  the  mixing  of  two  liquids  neither  of  which  is  itself 
"crystallin"  in  the  pure  condition.  The  mixture  therefore 


VISCOSITY  AND  FLUIDITY  97 

resembles  an  emulsion.  Bose  regards  all  "crystallin"  liquids  as 
emulsions  of  very  long  life,  i.e.,  they  settle  out  with  extreme 
slowness,  and  he  proposes  an  extension  of  the  kinetic  theory  to 
account  for  them.  According  to  van  der  Waals,  the  molecules 
are  to  be  regarded  as  spheres;  however,  the  molecules  of  sub- 
stances known  to  form  crystallin  liquids  do  not  approximate 
to  a  spherical  form  but  consist  of  two  benzene  rings  united  in 
such  a  way  as  to  make  a  rather  elongated  molecule.  Hence,  Bose 
thinks  that  they  may  be  better  represented  by  ellipsoids  of 
revolution.  As  the  temperature  is  lowered,  these  molecules 
naturally  arrange  themselves  with  their  long  axes  in  parallel 
planes.  As  the  molecules  unite  to  form  the  so-called  "swarms," 
the  viscosity  is  increased.  This  orderly  arrangement  also 
causes  the  liquids  to  show  double  refraction. 

It  was  shown  that  quite  often  the  viscosity  increases  rapidly 
as  the  temperature  is  raised  at  the  clarifying  point,  but  there  is 
also  then  an  increase  in  the  density. 

It  occurred  to  Bose,  Willers,  and  Rauert  (1909)  that  the 
orderly  "swarm"  arrangement  might  be  destroyed  by  measuring 
the  viscosity  under  conditions  for  turbulent  flow.  It  was  shown 
by  them  in  fact,  that  the  abnormalities  at  the  critical-solution 
temperature  do  decrease  as  the  transpiration  velocity  is  increased. 
But  these  results  are  not  very  conclusive  since  the  measurement 
of  viscosity  under  conditions  for  turbulent  flow  has  been  but 
little  investigated.  Pure  liquids  were  studied  by  them  under 
conditions  for  turbulent  flow  and  it  was  found  that  there  is  not  a 
complete  parallelism  between  the  viscosities  as  measured  by  the 
two  methods.  In  fact,  there  are  several  cases  where  one  sub- 
stance has  a  higher  viscosity  than  another  substance  under 
conditions  for  linear  flow,  but  a  lower  apparent  viscosity  under 
conditions  for  turbulent  flow.  No  explanation  seems  to  have 
been  given  for  this. 

Tsakalotos  (1910)  has  studied  mixtures  which  show  a  lower 
critical-solution  temperature,  triethylamine  and  water,  and 
nicotine  and  water,  as  well  as  amylen  and  aniline,  and  isobutyric 
acid  and  water.  He  used  only  one  or  two  temperatures  so  that 
the  peculiarity  with  which  we  are  here  concerned  did  not  appear. 

Bingham  and  White  (1911)  investigated  phenol  and  water 
mixtures  with  the  following  results.  (1)  The  fluidity  decreases 


98  FLUIDITY  AND  PLASTICITY 

unusually  rapidly  as  the  solutions  are  cooled  toward  the  critical- 
solution  temperature.  (2)  But  this  abnormality  appears  before 
the  critical-solution  is  reached  and  continues  on  and  through  the 
critical-solution  temperature.  (3)  In  the  region  where  the 
abnormality  appears,  it  is  very  difficult  to  obtain  concordant 
values  for  the  apparent  fluidity.  It  may  be  added  that  this  is  to 
be  expected  since  according  to  the  theory,  the  apparent  fluidity 
depends  upon  the  size  of  the  drops.  (4)  By  reflected  light  the  solu- 
tions in  this  region  appear  opalescent  :  by  direct  light  the  liquid 
shows  unequal  refraction,  the  images  of  objects  being  distorted. 

Drapier  (1911)  studied  two  mixtures  in  which  water  is  not 
a  component,  viz.  hexane  and  nitrobenzene,  and  cyclohexane  and 
aniline.  The  fluidity-temperature  curves  and  the  fluidity-weight 
concentration  curves  of  the  latter  are  shown  in  Fig.  37.  Drapier 
states  that  the  relations  are  similar  when  volume-concentrations 
are  employed.  According  to  his  experiments,  the  contention 
that  fluidities  are  normally  additive  in  homogeneous  mixtures  is 
fully  sustained. 

"II  semble  done  que  dans  un  intervalle  assez  etendu  de  varia- 
tion de  temperature  on  puisse  considerer  la  fluidite  comme  une 
fonction  line"aire  de  la  temperature,  sauf  pour  les  corps  tres 
associes  comme  1'eau,  ...... 

"  Pour  les  melanges,  loin  de  la  temperature  critique  la  variation 
de  la  fluidite  est  encore  lineaire.  Mais  plus  on  approche  de  la 
region  critique,  moins  les  formules  lineaires  sont  exactes.  Elles 
ne  peuvent  meme  plus  pretendre  a  un  semblant  d'exactitude, 
ainsi  que  le  montrent  bien  les  lignes  de  fluidite  des  melanges  a 
concentrations  voisines  de  la  concentration  critique:  elles  sont 
tout  a  fait  courbes  et  concaves  vers  1'axe  des  temperatures. 
D'ailleurs,  dej  pour  des  concentrations  eioignees  de  la  concentra- 
tion critique,  au  voisinage  de  la  temperature  de  demixtion  le  coeffi- 
cient de  fluidite  varie  tres  fort.  Mais  le  changement  est  plus 
graduel  pres  de  la  concentration  critique. 

"Si  1'on  examine  1'allure  des  isothermes  de  fluidite,  on  voit 
que  pour  les  melanges  de  corps  normaux  la  loi  d'additivite  : 


est  assez  bien  satisfaite  a  des  temperatures  superieures  a  la 
temperature  critique  de  dissolution.     J'ai  porte  en  abscisses  les 


VISCOSITY  AND  FLUIDITY 


99 


concentrations  en  poids,  mais  en  prenant  les  concentrations  en 
volume  la  loi  d'additivite"  n'est  pas  mieux  verifie*e.     Ce  n'est  que 


dans  le  voisinage  de  la  temperature  critique  qu'il  se  pre"sente  des 
hearts  singuliers,  resultant  de  la  courbure  des  lignes  d'egale 
concentration  et  se  traduisant  par  une  double  inflexion  des 
isothermes,  .  .  ." 


100  FLUIDITY  AND  PLASTICITY 

Commenting  on  the  theory  of  v.  Smoluchowski  by  way  of 
explanation  he  remarks,  "  II  est  probable  que  de  pareilles  he"tero- 
gene"ites  produirairent  une  augmentation  de  la  viscosite"  et 
pourraient  done  expliquer  la  courbure,  toujours  de  meme  sens, 
des  courbes  d'e"gale  concentration  et  par  consequent  les  ecarts  a 

101  d'additivite." 

These  researches  make  it  perfectly  clear  that  there  is  a  decrease 
in  the  fluidity  near  the  critical-solution  temperature  as  predicted 
and  that  in  some  way  this  decrease  is  connected  with  the  dis- 
appearance of  homogeneity  in  the  mixture.  Most  of  the  in- 
vestigators have  concerned  themselves  with  the  explanation  of 
disappearance  of  homogeneity  before  the  critical-solution  tem- 
perature is  reached,  rather  than  of  the  increase  in  viscosity. 


FIG.  38. — Diagram  illustrating  the  flow  of  emulsions. 

But  we  are  here  only  interested  in  the  fact  that  heterogeneity 
does  occur  simultaneously  with  the  abnormal  increase  in  viscosity, 
and  not  in  the  cause1  of  the  heterogeneity  itself. 

Scarpa  and  Bose  however  offered  explanations  of  the  abnormal 
increase  in  the  viscosity.  In  regard  to  Scarpa's  assumption 
that  the  decrease  in  fluidity  is  due  to  the  formation  of  hydrates, 
it  is  very  possible  that  hydrates  are  formed  between  phenol  and 
water,  with  which  he  worked;  but  he  has  not  given  any  facts  to 
prove  that  the  hydration  suddenly  increases  as  the  critical- 
solution  temperature  is  approached  even  in  this  favorable 
case.  In  the  cases  studied  by  Drapier  (cf.  Fig.  34),  such  a 
chemical  action  seems  to  be  out  of  the  question,  because  if  solva- 
tion  occurred  the  fluidity-concentration  curves  would  be  sagged 
even  above  the  critical-solution  temperature. 

In  order  to  understand  the  explanation  of  Bose,  we  refer  to 
Fig.  38  which  may  be  taken  to  represent  the  hypothetical 

1  For  an  attempted  explanation  cf.  Am.  Chem.  J.,  33,  1273  (1911). 


VISCOSITY  AND  FLUIDITY  101 

appearance  of  the  drops  of  an  emulsion  as  they  pass  through  a 
capillary  tube.  Due  to  the  friction  against  the  walls,  the  rear 
end  of  each  drop  is  flattened  and  the  front  end  is  unusually  convex. 
It  is  to  be  especially  noted  that  when  the  drops  are  small  in 
diameter  as  compared  with  the  diameter  of  the  tube  and  yet 
large  enough  to  occupy  the  whole  cross-section  of  the  tube, 
the  motion  of  the  liquid  is  by  no  means  entirely  linear,  being 
transverse  as  well  as  horizontal  as  indicated  by  the  arrows.  The 
effect  of  this  transverse  motion  is  to  increase  the  apparent 
viscosity  of  the  liquid.  If,  however,  the  drops  are  very  large  in 
comparison  to  the  diameter  of  the  tube,  the  importance  of  this 
transverse  motion  may  become  vanishingly  small.  Thus  if  the 
drops  of  an  emulsion  are  large  enough  to  fill  the  cross-section  of  a 
tube,  the  viscosity,  as  measured  by  the  rate  of  efflux,  will  be  at 
least  as  great  as  the  sum  of  the  component  viscosities,  but  it 
may  be  greater  due  to  the  transverse  motions.  We  grant  that 
below  the  critical-solution  temperature  a  part  of  the  increase  in 
viscosity  may  be  due  to  these  transverse  motions,  but  Bose 
would  seem  to  account  for  all  of  the  abnormal  increase  in  the 
viscosity  in  this  way.  This  however  is  not  warranted,  for  the 
reason  that  at  the  center  of  the  capillary  the  liquid  has  normally 
a  high  velocity  while  at  the  boundary  the  velocity  is  zero,  so  that 
there  is  a  considerable  tendency  for  any  drops  to  become  dis- 
rupted and  drawn  out  into  long  threads.  It  is  impossible  to 
believe  that  above  the  critical-solution  temperature  the  surface 
tension  of  the  " drops"  is  sufficient  to  prevent  disruption,  for 
we  are  accustomed  to  think  that  the  surface  tension  at  the  critical 
temperature  is  zero,  and  the  abnormality  in  the  fluidity  is  a 
maximum  at  this  temperature.  We  conclude  therefore  that 
neither  the  explanation  of  Scarpa  nor  of  Bose  is  sufficient,  but 
that  the  explanation  based  upon  the  nature  of  viscous  flow  in 
a  heterogeneous  mixture  is  both  necessary  and  sufficient. 

The  theory  requires  that  if  the  fluidities  of  the  two  components 
of  the  mixture  are  identical,  it  makes  no  difference  whether 
fluidities  or  viscosities  be  considered  additive;  hence  there  should 
be  no  irregularity  in  the  fluidity  curves  of  such  a  pair  of  sub- 
stances even  in  the  vicinity  of  the  critical-solution  temperature. 
No  case  has  been  examined,  so  far  as  we  know,  in  which 
the  components  have  approximately  the  same  fluidity  and 


102  FLUIDITY  AND  PLASTICITY 

the  mixture  has  a  critical-solution  temperature.  The  nearest 
approximation  is  in  the  case  of  isobutyric  acid,  <p2o°  =  76.0,  and 
water,  <p20°  =  99.8,  examined  by  Friedlander.  As  can  be  seen 
from  Fig.  39  taken  from  the  work  of  Drapier,  the  irregularity  is 
very  slight.  The  calculated  deviation  is 

an  +  6<p2 _|    ,  (36) 

aiji  +    6772 

which  for  a  50  per  cent  mixture  corresponds  to  27.6.  The  irregu- 
larity is  greatest  in  the  case  of  hexane,  ^20°  =  314.0,  and  nitro- 
benzene, <p2o°  =  50.1,  where  the  fluidities  are  also  the  most 
unequal.  The  calculated  deviation  is  in  this  case  95.6  for  a 
50  per  cent  mixture.  The  deviations  actually  read  from  the 
curves  for  isobutyric  acid  and  water  and  hexane  and  nitro- 
benzene are  of  the  order  of  11  and  25  respectively.  That  these 
numbers  are  so  much  smaller  than  the  calculated  values  may  be 
easily  accounted  for  on  the  supposition  that  the  drops  are  not  all 
sufficiently  large  to  fill  the  cross-section  of  the  capillary,  and 
hence  the  viscosities  are  not  strictly  additive. 

SUSPENSIONS 

According  to  the  view  that  viscosities  are  always  additive, 
the  viscosity  of  all  suspensions  should  be  infinite.  On  the  con- 
trary, as  already  stated,  Friedlander  found  that  colophonium 
suspended  in  water  had  practically  no  influence  on  the  viscosity 
of  water.  Similarly  Bose  measured  the  viscosity  of  suspensions 
of  finely-divided  quartz,  whose  viscosity  may  be  taken  as  infinite, 
in  bromoform  and  water,  and  he  found  that  the  viscosity  of  the 
medium  was  but  little  changed.  Had  they  measured  the  visco- 
sities at  increasing  concentrations  of  the  solid,  they  would  have 
undoubtedly  found  that  the  viscosity  was  altered  and  in  a 
perfectly  definite  manner.  As  already  indicated,  page  55  and 
Fig.  19,  the  fluidity  curves  of  such  suspensions  are  normally 
linear.  Since,  however,  suspensions  and  emulsions  are  closely 
allied,  it  is  important  to  inquire  why  viscosities  are  not  additive 
in  suspensions  as  well  as  in  the  emulsions  already  considered. 

In  suspensions,  we  have  practically  the  same  conditions  as  in 
emulsions  in  which  the  drops  are  so  small  that  they  do  not  nearly 
fill  a  cross-section  of  the  capillary  tube.  In  this  case  the  viscosi- 
ties are  not  strictly  additive. 


VISCOSITY  AND  FLUIDITY 


103 


104  FLUIDITY  AND  PLASTICITY 

If  for  simplicity  we  imagine  the  solid  particles  of  a  suspension 
to  be  all  united  into  sheets  parallel  to  the  direction  of  flow,  as  the 
shaded  areas  in  Fig.  33  then  it  is  evident  that  the  flow  will  be  the 
sum  of  the  flows  of  the  unshaded  areas,  i.e.,  the  fluidities  will  be 
strictly  additive  or 

since  <pi  is  practically  zero. 

But  if  these  solid  sheets  were  broken  up  into  fragments,  one  of 
which  is  shown  as  the  cross-section  of  a  cube  at  F  in  Fig.  40, 
the  deformation  of  the  liquid  would  tend  to  change  the  form  of 
the  cube  into  that  of  a  parallelepiped  as  shown  at  G,  but  as  the 
solid  is  rigid,  this  cannot  take  place;  so  that  the  shearing  force 
can  only  rotate  the  cube  around  its  center  as  shown  at  H.  But 
the  failure  of  the  solid  to  change  its 
shape  with  the  flow  of  the  liquid 
will  necessitate  transverse  motions 
in  the  liquid  by  way  of  readjust- 
ment, hence  the  viscosity  of  a 
suspension  will  always  be  greater 
than  it  would  be  were  the  fluidities 
strictly  additive.  If,  as  we  believe  can  be  proved  to  be  the 
case,  the  amount  of  transverse  motion  in  a  suspension  is  pro- 
portional to  the  number  of  suspended  particles  of  a  given  size, 
and  for  each  particle  the  amount  of  transverse  motion  bears  a 
constant  ratio  to  the  amount  of  shear,  it  will  follow  that  the 
fluidity  curves  of  suspensions  must  be  linear,  as  has  already 
been  shown  to  be  generally  true  (cf.  p.  55  and  Fig.  19). 

Enough  evidence  has  been  given  to  indicate  that  the  theory  of 
the  subject  and  the  most  diverse  sorts  of  experimental  data  are 
in  accord  in  supporting  the  fundamental  hypothesis  that  fluidities 
are  normally  additive  in  homogeneous  mixtures  and  fine  sus- 
pensions, but  not  in  heterogeneous  mixtures.  Much  additional 
evidence  could  be  given,  but  not  without  taking  up  subjects  out 
of  their  natural  order.  This  evidence  will  appear  as  we  proceed. 
For  further  confirmation  of  these  views  cf.  Drucker  and  Kassel 
(1911),  White  (1912). 

James  Kendall  (1913),  working  in  the  Nobel  Institute  of  Phys- 
ical Chemistry  under  the  direction  of  Professor  Arrhenius, 
concluded  that  "the  logarithmic  viscosity  (or  fluidity)  of  a  solu- 


VISCOSITY  AND  FLUIDITY  105 

tion  is  the  characteristic  additive  property,  and  not  these  quan- 
tities themselves."  This  conclusion  was  based  upon  data  which 
for  the  reasons  already  given  was  not  well  suited  for  reaching  a 
final  decision  of  the  matter.  As  the  result  of  more  recent  study 
of  the  matter  with  Monroe  (1919)  and  Wright  (1920)  Kendall  has 
come  to  the  conclusion  that  no  formula  tested  by  him  will  repro- 
duce the  observed  data.  The  present  author  is  in  hearty  accord 
with  this  conclusion  of  Kendall.  It  cannot  be  emphasized  too 
strongly,  to  the  novitiate  particularly,  that  no  single  formula  will 
reproduce  faithfully  any  considerable  portion  of  the  observed 
data  on  the  fluidity  of  mixtures.  Moreover  it  is  useless  to  look 
for  such  a  formula  in  the  present  state  of  our  knowledge.  A  much 
better  plan  is  to  assume  the  additivity  of  fluidities,  which  also 
has  the  virtue  of  being  the  simplest  hypothesis  that  we  can  make, 
and  then  try  to  account  for  the  deviations  from  the  exact  law 
on  the  basis  of  well-established  physical  and  chemical  evidence. 
If  the  fundamental  hypothesis  is  incorrect,  incongruities  will  soon 
develop  to  put  us  on  the  right  track.  If  correct,  we  should  pro- 
ceed as  rapidly  as  possible  to  exploit  the  new  knowledge  which 
fluidity  measurements  place  in  our  hands. 


CHAPTER  II 

FLUIDITY  AND  THE  CHEMICAL  COMPOSITION 
AND  CONSTITUTION  OF  PURE  LIQUIDS 

Attention  was  first  strongly  drawn  to  the  desirability  of  study- 
ing the  viscosity  of  homogeneous  liquids  in  relation  to  their  other 
properties  by  Graham  in  1861.  He  himself  measured  the 
viscosity  of  several  organic  liquids  at  the  uniform  temperature  of 
20°  and  noted  that  the  times  of  flow  increase  with  the  boiling- 
point,  from  which  he  inferred  that  there  is  a  connection  between 
viscosity  and  chemical  composition  similar  to  that  which  exists 
between  the  boiling-point  and  the  chemical  composition.  By 
comparing  the  times  of  flow  of  "equivalent  amounts,"  obtained 
by  multiplying  the  times  of  flow  of  equal  volume  by  the  molecu- 
lar weights  and  dividing  by  the  density  (rjAf/p),  Rellstab  (1868) 
sought  to  gain  a  more  intimate  knowledge  of  this  relation.  He 
measured  the  viscosity  over  a  range  of  temperatures  from  10  to 
50°  and  then  compared  the  substances  at  temperatures  at  which 
their  vapor-pressures  are  equal,  as  well  as  at  a  given  temperature. 
No  simple  quantitative  relationship  was  found  between  his  times 
of  flow  and  the  molecular  weight  or  vapor-pressure,  but  he  stated 
several  qualitative  relationships.  Thus  he  noted  that  the  time 
of  flow  always  decreases  as  the  temperature  rises,  that  an  incre- 
ment of  CH2  in  a  homologous  series  is  in  general  accompanied 
by  an  increase  in  the  time  of  flow,  but  that  metameric  substances 
may  have  very  different  efflux-times.  Without  attempting  a 
complete  summary  of  his  observations,  the  above  suffice  to  show 
that  he  regarded  temperature,  chemical  composition  and  con- 
stitution as  all  important  in  determining  the  rate  of  flow. 

Pribram  and  Handl  (1878-1881)  studied  a  large  number  of 
pure  liquids  over  a  range  of  temperatures  from  10  to  60°  express- 
ing their  results  in  "  specific  viscosities  "  taking  water  at  0°  as  100. 
Their  researches  marked  a  great  step  in  advance  but  only  to  the 
extent  of  confirming  and  extending  the  qualitative  observations 

106 


FLUIDITY  AND  THE  CHEMICAL  COMPOSITION         107 

of  Rellstab  and  also  of  Guerout,  and  not  in  establishing  quantita- 
tive relationships. 

Struck  by  the  fact  that  metameric  substances  sometimes  have 
such  widely  different  viscosities,  e.g.,  isobutyl  alcohol  0.03906 
and  ethyl  ether  0.002345  at  20°,  Briihl  (1880)  noted  that  the  one 
with  the  higher  viscosity  generally  had  the  higher  boiling-point 
and  index  of  refraction.  But  to  this  observation  Gartenmeister 
(1890),  testing  a  large  number  of  substances  at  20°  or  over  a 
range  of  temperatures,  found  numerous  exceptions. 

It  was  at  this  point  that  Thorpe  and  Rodger  (1894)  decided 
to  make  an  intensive  study  of  the  whole  subject  of  the  relation 
between  the  viscosity  of  liquids  and  their  chemical  nature.  Their 
first  care  was  to  work  out  a  method  which  would  give  them  a  far 
greater  precision  of  measurement  than  had  been  obtained  by 
many  of  their  predecessors.  They  then  carefully  purified 
some  87  substances  and  measured  their  viscosities  from  0°C  to 
the  boiling-point  of  each  substance.  An  exhaustive  search  was 
then  made  for  a  basis  of  comparison  which  would  bring  out  the 
quantitative  connection  between  the  viscosity  and  the  chemical 
nature  of  the  liquids.  In  this  search  they  compared  the  viscosity 
coefficients  (r;),  the  "molecular  viscosities"  [i?(Af/p)%],  the 
"molecular  viscosity  work"  (r^M/p),  and  in  order  to  make  the 
comparison  under  comparable  conditions  they  made  the  com- 
parisons at  the  boiling  temperatures,  at  "corresponding" 
temperatures,  at  temperatures  where  the  slopes  of  the  viscosity- 
temperature  curves  are  equal,  and  at  slopes  varying  under  speci- 
fied conditions.  They  furthermore  compared  the  constants  in 
the  empirical  equations  which  they  found  to  best  reproduce  the 
observed  viscosities  as  a  function  of  the  temperature,  and  they 
also  compared  the  temperatures  corresponding  to  a  given  slope 
in  the  viscosity-temperature  curves.  Their  choice  of  tempera- 
tures of  equal  slope  as  a  basis  of  comparison  deserves  a  word  in 
explanation.  They  found  that  on  comparing  the  viscosity 
curves  of  substances  which  gave  the  best  physico-chemical  rela- 
tionships at  the  boiling-point  that  the  general  shape  of  these 
curves  was  the  same,  or  in  other  words  the  slopes  of  the  substances 
at  their  boiling-points  were  practically  identical.  On  the  other 
hand,  alcohols  and  other  substances,  which  gave  little  evidence 
of  physico-chemical  relationships,  had  invariably  a  different 


108  FLUIDITY  AND  PLASTICITY 

slope.  It  therefore  occurred  to  them  to  compare  their  substances 
at  temperatures  of  equal  slope  and  they  seemed  to  find  theoret- 
ical justification  in  this  proceeding,  since  at  a  given  slope  the 
temperature  is  exercising  the  same  effect  upon  the  viscosity  of 
different  substances,  i.e.,  dri/dt  is  constant. 

They  were  able  to  establish  the  most  nearly  quantitative 
relationship  in  the  comparison  between  molecular  viscosity  work 
and  chemical  composition  and  constitution  using  a  constant 
slope,  arbitrarily  selected  as  0.000,032,3.  We  shall  now  examine 
the  nature  of  this  relationship. 

By  comparing  the  values  for  the  homologues  given  in  Table 
XXII  they  observed  that  the  addition  of  a  methylene  group 
to  a  compound  increases  the  observed  value  of  the  molecular 
viscosity  work  by  (80  +  5)  X  10~3  c.g.s.  units.  They  assume 
that  CH2  =  80,  the  factor  10~3  being  understood.  Similarly  an 
iso-grouping  is  found  to  lower  the  value  observed  for  the  normal 
compound  by  8  +  3  provided  that  the  highly  associated  butyric 
acids  are  left  out  of  the  calculation. 

The  value  of  H2  was  found  by  subtracting  the  value  of  nCH2, 
as  calculated  from  the  above  constants,  from  the  observed 
values  of  the  paraffins  whose  general  formula  is  CriH2n  +  2  as 
shown  in  Table  XXIII.  The  mean  value  of  H2  is  —68  and  since 
CH2  =  80,  C  =  148. 

Comparing  normal  propyl  with  allyl  compounds,  it  was 
found  that  the  occurrence  of  a  double  linkage  and  the  loss  of  two 
hydrogen  atoms  lower  the  molecular  viscosity  work  by  27  ±  1 ; 
hence  the  value  of  a  double  linkage  was  assumed  to  be  —95. 

Using  the  values  thus  obtained,  they  determined  the  value 
of  oxygen  in  ketones  to  be  — 19,  excluding  acetic  aldehyde  and 
dimethyl  ketone  from  the  calculation  because  they  are  the  first 
members  of  their  respective  series,  and  are  probably  associated. 
In  the  aliphatic  acids  the  two  oxygen  atoms  have  a  value  of 
81  +  4,  but  since  one  of  these  is  a  carbonyl  oxygen,  the  value  of 
hydroxyl  oxygen  must  be  100.  On  the  other  hand,  oxygen  when 
united  as  in  ether  was  found  to  be  43.  It  seemed  to  them  possible 
that  oxygen  might  have  yet  other  values  such  as  the  carbonyl 
oxygen  in  aldehydes  as  distinguished  from  ketones.  Comment- 
ing on  the  different  values  which  it  seemed  necessary  to  give  to 
the  same  atom  in  differently  constituted  compounds,  Thorpe 


FLUIDITY  AND  THE  CHEMICAL  COMPOSITION 


109 


TABLE  XXII.  —  MOLECULAR  VISCOSITY  WORK 

A  SLOPE  OF  0.000,032,3 


IN  ERGS  X  10~3  AT 


Substance 

Observed 

Difference 

Calculated 

Difference, 
per  cent 

Pentane  

329 

86.0 

332 

-0.9 

Hexane  

415 

80.0 

412 

0.7 

Heptane  

495 

79.0 

492 

0.6 

Octane  

574 

572 

0.3 

Isopentane  
Isohexane  

320 

404 

84.0 
78.0 

324 
404 

-1.2 
0.0 

Isoheptane  

482 

484 

-0.4 

Isoprene  

284 

72.0 

278 

2.1 

Diallyl  

356 

358 

-0.5 

Methyl  iodide  

255 

86.0 

264 

-3.5 

Ethyl  iodide 

311 

84.0 

344 

—0  9 

Propyl  iodide  

425 

424 

0.2 

I                1  '  d'd 

417 

88.0 

416 

0.2 

Isobutyl  iodide 

505 

496 

1.8 

Allyl  iodide 

399 

397 

0.5 

Ethyl  bromide  
Propyl  bromide  

282 
353 

71.0 

277 
357 

1.8 
-1.1 

Isopropyl  bromide  
Isobutyl  bromide  

346 
433 

87.0 

349 

427 

1.4 

Allyl  bromide  

327 

330 

-0.9 

Ethylene  bromide  
Propylene  bromide  

450 

526 

76.0 
88.0 

456 
536 

-1.3 
-1.9 

Isobutylene  bromide  

614 

608 

1.0 

Acetylene  bromide  

418 

409 

2.0 

Propyl  chloride  

294 

295 

-0.3 

Isopropyl  chloride  
Isobutyl  chloride  

290 
364 

74.0 

287 
367 

1.0 

-0.8 

Allyl  chloride  

268 

268 

0.0 

Methylene  chloride  
Ethylene  chloride  

241 
326 

85.0 
76.5 

244 
324 

-1.2 
0.6 

Methyl  sulfide  
Ethyl  sulfide  

240 

236 
396 

1.7 
-0.8 

Methyl  ethyl  ketone  
Methyl  propyl  ketone  
Diethyl  ketone  

302 
383 
376 

81.0 

301 
381 
381 

0.3 
0.5 
-1.3 

160 

77.0 

159 

0.6 

Acetic  acid  
Propionic  acid 

237 
323 

86.0 
74.0 

239 
319 

-0.8 
1.2 

Butyric  acid  

397 

399 

-0.5 

Isobutyric  acid 

398 

391 

1.8 

Acetic  anhydride  

394 

74.0 

0.3 

Propionic  anhydride 

542 

553 

-2.0 

Ethyl  ether  

295 

295 

0.0 

Benzene 

314 

81.0 

315 

-0.3 

Toluene 

395 

80.0 

395 

0.0 

Ethyl  benzene  
Ortho-xylene 

475 
483 

475 
475 

0.0 

1.7 

Meta-xylene  
Para-xylene  

474 
467 

475 
475 

-0.2 

-1.7 

110  FLUIDITY  AND  PLASTICITY 

TABLE  XXIII. — THE  VALUE  OF  HYDROGEN 


Substance 

n 

CnzHn+2 

nCH2 

H2 

5 

329 

400 

-71 

Normal  paraffins 

6 

7 

415 
495 

480 
560 

-65 
-65 

8 

574 

640 

-66 

5 

320 

392 

-72 

Iso-paraffins 

6 

405 

472 

-67 

7 

482 

552 

-70 

and  Rodger  remark  (p.  643),  "If  such  differences  are  confirmed 
by  more  numerous  observations,  viscosity  will  rank  as  one  of  the 
most  useful  properties  in  determining  the  constitution  of  oxygen 
compounds."  They  then  add,  "It  is,  of  course,  to  be  remem- 
bered here  that  the  value  of  hydroxyl  oxygen  as  it  is  derived  from 
the  acids  is  no  doubt  affected  by  molecular  complexity." 

Using  the  constants  obtained  as  above,  and  grouped  together 
in  Table  XXIV  for  reference,  Thorpe  and  Rodger  calculated  the 
values  of  the  molecular  viscosity  work  for  the  substances  given 
in  Table  XXII,  and  reproduced  in  column  4.  The  average  dif- 
ference between  the  observed  and  calculated  values  is  less  than 
1  per  cent,  but  it  is  to  be  remarked  that  water  and  the  alcohols 
do  not  enter  into  comparison  at  this  particular  slope.  At  a  differ- 
ent slope  they  were  able  to  bring  these  substances  into  the 
comparison,  and  they  found  a  very  great  divergence  between 
the  observed  and  calculated  values  amounting  to  44  per  cent  in 
the  case  of  dimethyl  ethyl  carbinol  and  47  per  cent  in  that  of 
water.  Again  the  difference  was  partly  attributable  to  constitu- 
tive influences,  since  it  was  noted  that  the  divergence  is  least  in 
the  primary  and  greatest  in  the  tertiary  alcohols.  But  at  the 
same  time  they  note  that  these  compounds  are  most  certainly 
associated  and  the  theoretical  values  of  the  molecular  weight  were 
used  in  place  of  the  actual  values.  They  conclude  their  study 
of  molecular  viscosity  work  at  equal  slope  with  the  following 
noteworthy  statement:  "The  results  here  obtained  are  of 
precisely  the  same  nature  as  those  discussed  under  molecular 
viscosity.  More  detail  has  been  given  to  show  that  the  sub- 
stances which  give  deviations  from  the  calculated  values  fall 


FLUIDITY  AND  THE  CHEMICAL  COMPOSITION         111 

into  two  classes.  In  the  first  the  deviations  are  to  be  attributed 
to  chemical  constitution,  as  similar  disturbing  effects  may  be 
detected  in  the  magnitudes  of  other  physical  properties  which  do 
not  seem  to  be  affected  by  molecular  complexity.  In  the  second 
are  those  substances  like  the  acids,  water,  and  the  alcohols,  for 
which  the  disturbing  factor  is,  no  doubt,  molecular  complexity, 
the  effect  produced  in  this  way,  in  the  case  of  the  alcohols,  being 
dependent  upon  their  chemical  nature."  Thorpe  and  Rodger 
have  done  great  service  in  stating  the  problem  before  us  so 
clearly.  At  a  subsequent  point  in  our  discussion,  we  will  show 
how  by  a  different  method  of  comparison  it  is  possible  to  largely 
avoid  the  first  cause  of  discrepancy  given  above,  and  how  then 
with  only  one  unknown  quantity  remaining,  it  is  possible  to  get  a 
proximate  solution  of  the  problem. 

TABLE  XXIV. — MOLECULAR  VISCOSITY  WORK  CONSTANTS  AT  SLOPE 
0.000,032,3 

Hydrogen —  34 

Carbon 148 

Hydroxyl-oxygen,  C-O-H 100 

Ether-oxygen,  C-O-C 43 

Carbonyl-oxygen,  C  =  O —   19 

Sulfur,  C-S-C 144 

Chlorine  (in  monochlorides) 89 

Chlorine  (in  dichlorides) 82 

Bromine  (in  monobromides) 151 

Bromine  (in  dibromides) 148 

Iodine 218 

Iso-grouping —     8 

Double  linkage —  95 

Ring-grouping —369 

The  effect  of  chemical  constitution  upon  viscosity  has  been 
employed  to  good  effect  in  the  solution  of  several  much-mooted 
chemical  problems  by  Dunstan  and  Thole  and  their  co-workers. 
Thus  Thole  (1910)  observed  a  steady  increase  in  the  viscosity  of 
freshly  distilled  ethyl  acetoacetate  owing  to  the  gradual  enoliza- 
tion  of  the  ketonic  form.  Hilditch  and  Dunstan  (1911)  have 
observed  that  the  presence  of  Thiele's  "conjugated  double  bonds" 
in  compounds  produces  a  great  increase  in  the  viscosity.  Thole 
(1912)  has  shown  that  the  viscosity  method  can  be  used  to  dis- 
tinguish between  geometrical  isomerides  like  maleic  and  fumaric 


112  FLUIDITY  AND  PLASTICITY 

acids.  But  while  they  attribute  these  effects  to  the  constitution 
of  the  molecules,  it  should  be  noted  that  the  immediate  cause  of 
the  increase  in  viscosity  may  in  each  case  be  association,  which 
is  the  same  as  saying  that  it  may  be  due  to  chemical  composition 
as  distinguished  from  chemical  constitution.  Certainly  com- 
pounds containing  the  hydroxyl  radical  are  often  associated  and 
these  same  compounds  are  noted  for  their  high  viscosity,  so  that 
in  the  case  of  ethyl  acetoacetate  the  way  seems  open  to  explain 
the  greater  viscosity  of  the  enol  form  on  the  basis  of  an  associated 
molecule,  quite  as  well  as  on  the  basis  of  symmetry  or  other 
constitutive  influence.  At  first  sight  it  seems  as  though  consti- 
tutive influences  must  solely  and  immediately  determine  the 
viscosity  values  in  each  of  the  above  examples,  but  Thole  (1912) 
seems  to  realize  that  this  is  not  the  actual  case  with  maleic  and 
fumaric  acids,  the  latter  of  which  gives  the  higher  viscosity  in 
methyl  alcohol  solution.  He  says,  "  The  viscosities  of  the  isomers 
depend  not  only  on  the  relative  positions  of  the  unsaturated 
groups  but  also  on  the  degree  of  residual  affinity"  which  causes 
molecular  association.  Thus  the  "adjacent"  maleic  acid  may 
have  the  lower  viscosity  due  to  slighter  association.  This  view 
is  borne  out  by  the  fact  noted  by  Thole  that  "barium  fumarate 
crystallizes  with  three  molecules  of  water  while  barium  maleate, 
in  which  the  residual  affinities  of  the  carboxyl  groups  are  more 
nearly  mutually  satisfied,  combines  with  only  one  molecule  of 
water." 

To  what  extent  different  constitutive  influences  affect  the 
association  of  compounds  is  an  exceedingly  important  subject 
but  it  is  not  relevant  to  our  discussion  of  viscosity.  Our  problem 
is  to  study  the  immediate  effects  of  constitutive  influences  and 
the  chemical  composition  of  the  molecule  upon  the  viscosity  and 
to  estimate  their  relative  importance. 

Regardless  of  how  much  uncertainty  there  may  be  in  regard 
to  the  importance  of  constitutive  influences  on  viscosity,  there 
can  be  no  doubt  about  the  importance  of  chemical  composition. 
All  evidence  shows  that  this  factor  is  of  great  importance.  Dun- 
stan  and  Langton  (1912)  have  made  use  of  this  for  the  determina- 
tion of  transition  points,  and  Thole  (1913)  in  the  detection  of  the 
presence  of  racemic  compounds  in  the  liquid  state,  and  many  other 
instances  might  be  cited. 


FLUIDITY  AND  THE  CHEMICAL  COMPOSITION 


113 


Comparison  of  Fluidities. — We  have  already  given  reasons  for 
believing  that  if  liquids  were  completely  unassociated  and 
expanded  in  a  linear  manner  with  the  temperature,  the  fluidity- 
temperature  curves  would  be  straight  lines.  To  compare  a 
family  of  curves  which  are  straight  lines  is  a  simpler  task  than 
the  comparison  of  a  family  of  hyperbolas,  hence  it  seems  a  justi- 


FIG.  41.— The  fluidities  of  vari- 
ous hydrocarbons  at  different 
temperatures  and  extrapolated 
to  their  boiling  temperatures. 
4.  Pentane;  5.  Isopentane;  6. 
Hexane;  1.  Isohexane;  8.  Hep- 
tane; 9.  Isoheptane;  10.  Octane; 
11.  Trimethylethylene;  12.  Iso- 
prene;  13.  Diallyl;  56.  Benzene; 
57.  Toluene;  58.  Ethylbenzene; 
59.  (o)-Xylene  60.  (m)-Xylene; 
61.  (p)-Xylene. 


7 


FIG.  42. — The  fluidities  of  various 
ethers  and  acid  anhydrides  at  differ- 
ent temperatures  and  extrapolated 
to  their  boiling  temperatures.  53. 
Acetic  anhydride;  54.  Propionic 
anhydride;  55.  Diethyl  ether;  83. 
Methyl  propyl  ether;  84.  Ethyl 
propyl  ether;  85.  Dipropyl  ether; 
86.  Methylisobutyl  ether;  87. 
Ethylisobutyl  ether. 


fiable  expectation  that  we  may  be  able  to  find  simpler  relations 
by  a  suitable  comparison  of  fluidities.  Before  deciding  on  a 
basis  of  comparison  let  us  inspect  the  fluidity-temperature 
curves  as  obtained  from  the  observations  of  Thorpe  and  Rodger 
as  given  in  Figs.  41  to  46.  Confining  our  attention  first  of  all 
to  the  aliphatic  hydrocarbons  in  Fig.  41  we  see  that  near  their 
boiling-points,  indicated  by  small  circles  in  the  figure,  the  fluidity 


114 


FLUIDITY  AND  PLASTICITY 


curves  are  nearly  straight  and  parallel  lines.  However  as  we  get 
away  from  the  boiling-temperature,  there  is  a  curvature  present 
so  that  it  is  probable  that  the  fluidity  curve  would  reach  the  tem- 
perature axis  asymptotically  as  the  temperature  were  lowered. 
Broadly  speaking,  the  curves  of  a  given  homologous  series  near 
their  boiling-points  consist  of  a  series  of  parallel  straight  lines, 
which  are  therefore  completely  denned  mathematically  by  their 
slopes  and  intercepts.  We  find  the  same  thing  in  other  series, 


y 


FIG.  43.— The  fluidities  of  vari- 
ous bromides  at  different  tem- 
peratures. 20.  Ethyl  bromide; 
21.  Propyl  bromide;  22.  Iso- 
propyl bromide;  23.  Isobutyl 
bromide;  24.  Allyl  bromide;  25. 
Ethylene  bromide;  26.  Propy- 
lene  bromide;  27.  Isobutylene 
bromide;  28.  Acetylene  bro- 
mide. 


Z5' 


100' 


FIG.  44. — The  fluidities  of  various 
iodides  at  different  temperatures. 
14.  Methyl  iodide;  15.  Ethyl  iodide; 
16.  Propyl  iodide;  17.  Isopropyl 
iodide;  18.  Isobutyl  iodide;  19. 
Allyl  iodide. 


as  the  ethers  and  acid  anhydrides  given  in  Fig.  42,  but  it  is  clear 
that  the  slope  is  different  in  the  two  classes.  The  slope  then  is 
dependent  upon  the  class  to  which  a  compound  belongs  and  the 
intercepts  are  dependent  upon  the  chemical  composition. 
According  to  this  broad  aspect  of  the  case  it  should  make  no  dif- 
ference whether  we  compare  fluidities  at  a  given  temperature  or 
temperatures  corresponding  to  a  given  fluidity.  But  there  are 
several  reasons  for  choosing  the  latter  basis  of  comparison  rather 
than  the  former. 


FLUIDITY  AND  THE  CHEMICAL  COMPOSITION 


115 


1.  The  slopes  of  the  fluidity-temperature  curves  for  a  given 
homologous  series  are  more  nearly  the  same  when  the  fluidities 
are  equal. 

2.  When  the  fluidities  are  the  same,  the  vapor-pressures  are 
nearly  equal,  and  experience  has  shown  that  substances  are 
comparable  at  temperatures  which  correspond  to  equal  vapor- 
pressure. 

3.  The  fluidity  curves  of  associated  substances  like  the  alcohols, 
Fig.  46,  depart  widely  from  linearity  at  low  fluidities,  although 
they  approach  linearity  at  high  fluidities,  as  do  the  curves  of 
other  compounds. 

4.  A  yet  more  cogent  reason  grows  out  of  the  fact  that  exact 


1*5- 

Fio.  45. — The  fluidities  of  various  organic  acids  at  different  temperatures. 
48.  Formic  acid;  49.  Acetic  acid;  50.  Propionic  acid;  51.  Butyric  acid;  52. 
Isobutyric  acid. 


parallelism  in  the  curves  of  a  given  class  is  not  to  be  expected 
since  all  fluidity-temperature  curves  must  undoubtedly  meet  at 
the  absolute  zero  of  temperature.  Hence  while  it  may  require 
a  constant  increment  of  temperature  to  produce  a  given  fluidity 
as  each  methylene  group  is  added  to  the  molecule,  it  is  absolutely 
certain  that  a  constant  decrement  of  the  fluidity  at  a  given  tem- 
perature cannot  be  expected  as  each  methylene  group  is  added. 
Thus  a  methylene  group  added  to  pentane,  Fig.  37,  lowers  the 
fluidity  at  0°  by  a  certain  amount,  but  the  effect  of  adding  a 


116 


FLUIDITY  AND  PLASTICITY 


methylene  group  to  heptane  is  less  and  the  effect  of  adding 
other  methylene  groups  must  be  still  less,  otherwise  it  would 
require  no  very  high  molecular  weight  to  give  a  negative  fluidity, 
which  is  inconceivable. 

.  The  fluidity  of  200  is  chosen  as  a  basis  of  comparison  in  order 
that  as  large  a  number  of  substances  as  possible  may  be  included. 
The  absolute  temperatures  and  slopes  of  several  unassociated 


SO" 


100" 


FIG.  46. — The  fluidities  of  various  alcohols  at  different  temperatures.  62. 
Methyl  alcohol;  63.  Ethyl  alcohol;  64.  Propyl  alcohol;  65.  Isopropyl  alcohol; 
66.  Butyl  alcohol;  67.  Isobutyl  alcohol;  68.  Trimethyl  carbinol;  69.  Active 
amyl  alcohol;  70.  Inactive  amyl  alcohol;  71.  Dimethylethylcarbinol ;  72. 
Allyl  alcohol. 


compounds  corresponding  to  the  fluidity  of  200  are  given  in 
Table  XXV.  The  third  column  of  this  table  shows  that  the 
value  of  a  methylene  grouping  varies  around  a  mean  value  of 
22.7,  the  mean  deviation  from  this  value  being  3.  The  effect 
of  an  iso-grouping  is  to  decrease  the  temperature  required  by 
about  7.6°,  as  shown  in  Table  XXVI. 


FLUIDITY  AND  THE  CHEMICAL  COMPOSITION 


117 


TABLE  XXV. — ABSOLUTE  TEMPERATURES  AND  SLOPES  OF  NON-ASSOCIATED 
SUBSTANCES  CORRESPONDING  TO  A  FLUIDITY  OP  200  C.G.S.  UNITS 


Substance 

Absolute 
tempera- 
ture (<f>  = 
200)   ob- 
served 

Differ- 
ence 
CH2 

Slope  at 
(</>  = 
200) 

Absolute 
tempera- 
ture (<t> 
=  200) 
calculated 

Per 

cent, 
differ- 
ence 

Hexane 

(255.  1)1  \ 

(2  .  88) 

254.6 

0.2 

Heptane  

276.1     J 

(21.0) 

277.3 

0.4 

Octane  

299.1 

23.0 

2.44 

300.0 

0.3 

Isohexane  

(249.0) 

(on  1\ 

(2.79) 

247.0 

0.8 

Isoheptane  

269.2 

\£\j  .  &) 

2.68 

269.7 

0.2 

Methyl  iodide  

290.2 

19.0 

1.92 

287.4 

1.0 

Ethyl  iodide  

309.2 

OO     K 

1.80 

310.1 

0.3 

Propyl  iodide  

332.7 

&6  .  0 

1.82 

332.8 

0.0 

Isopropyl  iodide  

324.5 

01    f) 

1.92 

325.2 

0.2 

Isobutyl  iodide  

345  .  5 

£t\.  .  U 

1.86 

347.9 

0.7 

Allyl  iodide  

330.5 

1.82 

328.8 

0.5 

Ethyl  bromide  

268.7 

27  9 

2.22 

273.5 

1.8 

Propyl  bromide  

296.6 

2.08 

296.2 

0.1 

Isopropyl  bromide  

289.4 

oc    a 

2.22 

273.5 

1.8 

Isobutyl  bromide  

315.0 

ZO  .  O 

2.08 

311.3 

1.1 

Ethyl  propyl  ether  

(255.0) 

(94  01 

(2.70) 

256.1 

0.5 

Dipropyl  ether 

279  .  0     f 

\6<t  .  u; 

2.62 

278.8 

0.1 

Methylisosbutyl  ether.  .  . 

(251.1)   1 

f-\  f\  f\\ 

(2.75) 

248.5 

1.0 

Ethylisobutyl  ether  

270.1     / 

Uy.uj 

2.68 

271.2 

0.4 

Values  in  parentheses  are  extrapolated. 

TABLE  XXVI. — THE  VALUE  OF  THE  ISO-GROUPING 


Temperature  ob- 

Temperature ob- 

Substance 

served,  normal 

served,  iso- 

Difference 

grouping 

grouping 

Hexane  

255.1 

249.0 

6.1 

Heptane  ... 

276.1 

269.2 

6.9 

Propyl  iodide  

332.7 

324.5 

8-2       „ 

Propyl  bromide  .  .  . 

296.6 

289.4 

7.2 

Propyl  chloride.  .  .  . 

261.5 

255.2 

6.3 

Butyric  acid  

381.6 

371.6 

10.0 

Methyl  butyrate  .  . 

304.2 

295.8 

8.4 

The  value  for  the  hydrogen  atom  is  calculated  as  follows : 


118  FLUIDITY  AND  PLASTICITY 

TABLE  XXVII. — THE  VALUE  OF  THE  HYDROGEN  ATOM 


Substance 

Temperature 
observed 

nCH2 
calculated 

Difference 

Hexane  

255.1 

136.2 

118.9 

Heptane  
Octane  

276.1 
299.1 

158.9 
181.6 

117.2 
117.5 

Isohexane  

249.0 

128.6 

120.4 

Isoheptane  

269.2 

151.3 

117.9 

The  value  for  H2  is  118.4  ±  1.0.     The  hydrogen  atom  has 
therefore  a  value  of  59.2  and  the  carbon  atom  of  —95.7. 

The  value  of  the  "double  bond"  in  allyl  compounds  is  obtained 
from  Table  XXVIII. 

TABLE  XXVIII.— THE  VALUE  OP  THE  DOUBLE  BOND 


Substance 

Temperature  ob- 
served, normal 
propyl 

Temperature  ob- 
served, allyl 

Difference 

Iodides  

332.7 

330.7 

2.2 

Bromides  

296.6 

292.2 

4.4 

Chlorides  

261.5 

256.0 

5.5 

To  raise  the  fluidity  of  an  allyl  compound  to  200  it  is  only 
necessary  to  raise  it  to  a  temperature  which  is  some  4°  lower  than 
is  necessary  for  the  corresponding  normal  compound,  containing 
two  more  hydrogen  atoms.  Thus  the  "double  bond"  has  a 
value  of  1 14.4,  the  absence  of  the  hydrogen  atoms  being  nearly 
compensated  for  by  the  "condition  of  unsaturation." 

Assuming  that  the  ethers  are  unassociated,  we  may  obtain  the 
value  of  the  oxygen  atom. 

TABLE  XXIX. — THE  VALUE  OF  THE  OXYGEN  ATOM 


Substance 

Temperature 
observed 

C»H2n  +  2           Oxygen 

Ethylpropyl  ether  
Dipropyl  ether  
Methylisobutyl  ether  
Ethylisobutyl  ether  

254.9 
279.0 
251.4 
270  3 

231.9 
254.6 
224.3 
247.0 

23.0 
24.4 
27.1 
23.3 

FLUIDITY  AND  THE  CHEMICAL  COMPOSITION         119 

This  gives  an  average  value  for  oxygen  of  24.2  with  an  average 
divergence  of  1.3  from  this  mean.  From  these  values,  the 
absolute  temperatures  corresponding  to  a  fluidity  of  200  may  be 
calculated.  Some  of  these  calculated  values  are  given  in  the 
fifth  column  of  Table  XXV.  A  comparison  between  these 
calculated  and  the  observed  values  for  35  substances  shows  an 
average  percentage  difference  of  less  than  0.8  per  cent. 

Association.— In  attempts  to  establish  a  relation  between 
viscosity  and  chemical  composition  it  has  been  customary  to 
disregard  entirely  the  fact  that  certain  classes  of  substances  are 
known  to  be  highly  associated,  and  hence  the  molecular  values  as 
calculated  from  the  atomic  constants  cannot  be  expected  to  agree 
with  the  observed  values.  A  more  logical  method  of  procedure 
would  be  to  use  known  values  of  the  association  in  arriving  at 
the  calculated  molecular  temperatures.  The  difficulty  of  this 
method  is  that  the  values  of  the  association  as  given  by  different 
methods  do  not  agree  very  closely  and  even  the  methods  of 
getting  these  values  have  been  subjected  to  criticism.  It  seems 
best  therefore  to  reverse  the  method  and  use  our  atomic  constants 
to  calculate  the  association,  which  can  then  be  compared  with  the 
values  of  the  association  obtained  from  the  surface  tension, 
volume,  et  cetera. 

In  the  calculation  of  the  atomic  constants  as  given  above, 
it  was  assumed  that  the  compounds  chosen  were  non-associated. 
This  is  not  entirely  warranted,  but  they  must  be  associated  to 
approximately  the  same  extent  since  the  agreement  between  the 
calculated  and  observed  values  is  generally  satisfactory,  and  it  is 
the  general  belief  that  some  of  these  compounds  are  indeed 
unassociated.  It  is  highly  probable  that  association  or  constitu- 
tion is  responsible,  in  part  at  least,  for  the  uncertainty  in  the 
so-called  "constants,"  but  this  uncertainty  can  be  removed  by 
further  amplification  of  our  data. 

Since  the  atomic  constants  are  additive,  it  follows  directly 
that  the  association  will  be  given  by  the  ratio  of  the  observed  to 
the  calculated  values  of  the  temperature  corresponding  to  the 
given  fluidity.  Thus  for  water  (H2O)Z  at  the  fluidity  of  200 
the  absolute  temperature  is  328.9,  while  the  value  calculated 
from  the  gas  formula  H20  is  2  X  59.2  +  24.2  =  142.6.  The 
association  factor  (x)  at  the  temperature  of  observation  (328.9° 


120 


FLUIDITY  AND  PLASTICITY 


absolute)  is  therefore  328.9/142.6  =  2.31.  In  Table  XXX  are 
given  the  observed  and  calculated  absolute  temperatures  corre- 
sponding to  the  fluidity  of  200  and  the  association  calculated 
therefrom  for  some  typical  associated  compounds.  The  slopes 
of  these  curves  are  also  given  in  the  fourth  column. 

TABLE  XXX. — ABSOLUTE   TEMPERATURES   AND   SLOPES  OF   SOME   ASSO- 
CIATED COMPOUNDS  CORRESPONDING  TO  A  FLUIDITY  OF  200  C.G.S.  UNITS 


Absolute 

Absolute 

Substance 

temperature 
for  (0  =  200) 

temperature 
for  (<t>  =  200) 

Slope  for 
(0  =  200) 

Association 

observed 

calculated 

Water  

328.9 

142.6 

3.04 

2.31 

Formic  acid  

(380.2) 

185.5 

(2.18) 

2.05 

Acetic  acid  

363.8 

208.2 

2.06 

1.77 

Propionic  acid  

362.0 

230.9 

1.92 

1.57 

Butyric  acid  

381.6 

253.6 

1.92 

1.57 

Isobutyric  acid 

371.6 

246.0 

2.00 

1.51 

Methyl  alcohol 

305.2 

165.3 

2.78 

1.84 

Ethyl  alcohol  

343.4 

188.0 

3.24 

1.83 

Propyl  alcohol  

365.6 

210.7 

3.76 

1.74 

Butyl  alcohol  

377.0 

233.4 

3.44 

1.62 

Ethyl  formate  

273.8 

230.7 

2.40 

1.19 

Ethyl  acetate  

284.0 

253.4 

2.50 

1.12 

Ethyl  propionate  

298.1 

275.1 

2.44 

1.08 

The  test  of  our  complete  process  of  reasoning  comes  now 
when  we  compare  the  association  obtained  in  this  way  with  the 
values  which  have  been  obtained  by  other  methods.  The  results 
of  this  comparison  are  shown  by  Table  XXXI. 

So  far  as  one  is  able  to  judge,  the  result  seems  to  be  all  that 
could  be  desired.  There  are  almost  invariably  values  given  by 
other  methods  which  are  both  higher  and  lower  than  our  values 
and  such  a  degree  of  association  is  certainly  not  inconsistent  with 
our  knowledge  of  the  chemical  conduct  of  these  substances. 
The  fluidity  method  of  obtaining  the  association  factor  seems  to 
be  freer  from  assumptions,  to  which  questions  maybe  raised,  than 
other  methods  which  have  been  proposed,  and  it  is  to  be  hoped 
that  it  may  prove  useful  in  calculating  this  very  important  fac- 
tor. If  eventually  we  are  able  to  obtain  thoroughly  consistent 


FLUIDITY  AND  THE  CHEMICAL  COMPOSITION 


121 


TABLE  XXXI. — A  COMPARISON  OF  THE  VALUES  OF  ASSOCIATION  AS  DETER- 
MINED BY  DIFFERENT  INVESTIGATORS 


Substance 

R.  &  S.,i 
16-46° 

R.  &  S., 
corrected 
by  Traube 

Traube,2 
15° 

Longi- 
nescu* 

B.  &  H.,<  tem- 
perature of  (<t> 
=  200) 

Water  

/3.55 
\1.64 

1.79 

3.06 

4.67 

2.31 

Dimethyl  ketone  

1.26 

1.18 

1.53 

1.60 

1.23 

Diethyl  ketone  

1.25 

1.16 

Methyl  propyl  ketone.  .  . 

1.11 

1.10 

1.43 

.25 

1.14 

Formic  acid  

3.61 

2.41 

1.80 

.80 

2.05 

Acetic  acid  

/3.62 
\2.13 

2.32 

1.56 

.75 

1.77 

Propionic  acid 

1.77 

1.45 

1.46 

.55 

1.57 

Butyric  acid 

1  .58 

1.35 

1.39 

.36 

1.51 

Isobutyric  acid 

1.45 

1.28 

1.31 

1.51 

Benzene  

1.01 

1.05 

1.18 

>1.17<1.31 

Toluene  

0.94 

1.01 

1.08 

>  1.  08  <  1.517 

Methyl  alcohol  

!3.  43 

2.53 

1.79 

3.17 

1.84 

2.32 

Ethyl  alcohol  

2.74 

1.80 

1.67 

2.11 

1.83 

1.65 

Propyl  alcohol  

2.25 

1.70 

1.66 

1.67 

1.74 

Isopropyl  alcohol 

2.86 

2.00 

1.53 

1.75 

Butyl  alcohol  

.94 

1.47 

1.62 

Isobutyl  alcohol  

.95 

1.53 

1.54 

1.66 

Active  amyl  alcohol  

.97 

1.54 

1.53 

1.54 

Allyl  alcohol  

.88 

1.50 

1.55 

1.80 

1.69 

Methyl  formate     

.06 

1.07 

(1.60) 

1.12 

1.25 

Ethyl  formate 

.07 

1.08 

1  .  39o° 

1.19 

Methyl  acetate  

.00 

1.04 

1.48o° 

1.09 

1.17 

Ethyl  acetate 

0.99 

1.04 

1.25 

1.00 

1.12 

Propyl  acetate 

0.92 

1.00 

1.31 

1.00 

1  .  11 

Ethyl  propionate  

0.92 

1.00 

1.27 

0.94 

1.08 

Methyl  butyrate  

0.92 

1.00 

1.30o° 

1.00 

1.10 

1  RAMSAY  and  SHIELDS,  Zeitschr.  f.  physik.  Chem.,  12,  464  (1893);  15,  115  (1894). 
'  TRAUBE,  Ber.  d.  deutsch.  chem.  Gesell.,  30,  273  (1897). 
»  /.  Mm.  Phys.,  1,  289  (1903). 
4  BINQHAM  and  HARBISON,  loc.  cit. 

results  from  the  different  methods,  it  is  interesting  to  observe 
that  it  should  be  possible  to  calculate  the  volume,  surface 
•tension,  et  cetera,  even  of  associated  liquids  from  their  atomic 
constants  and  their  fluidities. 

Fluidity  and  Chemical  Constitution. — Dunstan  and  Thole 
(Viscosity  of  Liquids,  page  31)  have  very  properly  called  attention 
to  the  fact  that  the  differences  between  the  calculated  and 
observed  values  of  the  fluidity  in  Table  XXV  "are  due  not  only 


122 


FLUIDITY  AND  PLASTICITY 


to  association  but  to  want  of  sufficient  data  for  calculating  accu- 
rately the  atomic  'constants'  and  also  to  constitutional  effects, 
such  as  the  mutual  influence  of  groupings  in  the  molecule,  sym- 
metry and  so  forth."  As  was  intimated  earlier  in  this  chapter, 
to  chemical  constitution  has  generally  been  attributed  a  very 
large  effect  on  viscosity,  but  it  often  turns  out  on  investigation 
that  this  supposed  constitutive  influence  occurs  in  substances 
that  are  known  to  be  associated  and  this  association  was  not  taken 
into  account,  and  in  other  cases  the  supposed  constitutive  influ- 
ence is  almost  certainly  purely  a  hypothesis  framed  to  explain 
an  unnoticed  defect  in  the  method  of  comparison.  We  shall  now 
give  some  facts  to  support  these  bare  statements  and  we  shall 
then  investigate  the  important  question  as  to  whether  this  dwind- 
ling constitutive  effect,  as  distinct  from  the  effect  of  association, 
can  safely  be  disregarded  altogether. 

In  assigning  values  to  the  halogen  atoms,  Thorpe  and  Rodger 
(p.  669  et  seq.)  found  it  necessary  to  give  a  different  value  to 
chlorine  in  monochlorides,  dichlorides,  trichlorides  and  tetra- 
chlorides,  but  even  then  the  results  are  not  satisfactory  since  in 
ethylene  and  ethylidene  chlorides  the  value  which  must  be 
assigned  the  chlorine  atom  is  certainly  different.  How  the  effect 
of  the  chlorine  atom  varies  at  the  fluidity  of  200  is  shown  in  the 
fourth  column  of  Table  XXXII. 

TABLE  XXXII. — THE  VALUE  OF  THE  CHLORINE  ATOM 


Substance 

Absolute  tem- 
perature (<f>  = 
200),  observed 

Hydro- 
carbon 
residue, 
calculated 

Chlorine 

Associa- 
tion 

Propyl  chloride  

261.5 

127.3 

134.2 

1.105 

Isopropyl  chloride  

255.2 

119.7 

135.5 

.11 

Isobutyl  chloride  

285.2 

142.4 

142.8 

.13 

Allyl  chloride  

256.0 

123.3 

132.7 

.10 

Ethylene  chloride  

336.5 

45.4 

145.5 

.27 

Ethylidene  chloride..  .  . 

291.2 

45.4 

122.9 

.10 

Methylene  chloride.  .  .  . 

279.1 

22.7 

128.7 

.15 

Chloroform  

305.3 

-  36.5 

113.9 

.04 

Carbon  tetrachloride  .  . 

347.0 

-  95.7 

110.7 

.01 

Carbon  dichloride  

356.3 

-  77.0 

108.3 

0.99 

FLUIDITY  AND  THE  CHEMICAL  COMPOSITION         123 

There  is  then  a  somewhat  regular  decrease  in  the  apparent  value 
of  chlorine  as  the  number  of  chlorine  atoms  in  the  molecule  are 
increased.  How  much  of  this  is  due  to  constitutive  influence 
directly  and  how  much  can  be  explained  on  the  ground  of  asso- 
ciation? Ramsay  and  Shields  and  Traube  agree  that  carbon 
tetrachloride  is  very  little  associated  if  at  all,  Ramsay  and 
Shields  giving  the  value  1.01  and  Traube  1.00i5°.  If  then  we  take 
the  average  of  the  closely  agreeing  values  of  the  two  compounds 
containing  four  chlorine  atoms  we  obtain  as  the  value  of  the 
chlorine  constant  109.5  and  with  this  we  can  calculate  the  asso- 
ciation of  the  other  compounds..  The  values  thus  obtained  are 
given  in  the  fifth  column  of  Table  XXXII.  Ethylene  chloride 
is  seen  according  to  this  method  of  calculation  to  be  highly 
associated,  but  Traube  has  given  a  still  higher  value  for  the  asso- 
ciation at  15°  of  1.46.  Data  for  the  other  chlorides  is  lacking, 
but  calculating  the  association  of  propyl  chloride  by  the  method 
of  Traube,  the  author  obtains  the  value  of  1.11  which  agrees 
excellently  with  our  value  of  1.105.  The  mono-halides  seem  to 
be  usually  associated  according  to  Traube  for  he  gives  for  methyl 
iodide  1.30,  for  ethyl  iodide  1.19  and  for  ethyl  bromide  1.28.  It 
is  greatly  to  be  regretted  that  our  available  data  is  so  meager, 
but  for  the  present  we  can  only  conclude  that  the  effect  of  con- 
stitution upon  the  value  of  the  chlorine  atom  is  too  small  to  be 
detected. 

In  reference  to  the  lack  of  constancy  in  the  value  of  a  methyl- 
ene  group  in  Table  XXV,  it  seemed  desirable  to  take  the  average 
of  as  large  number  of  values  as  possible,  but  with  the  limited 
data  on  hand  this  made  it  necessary  to  include  a  number  of 
compounds  which  are  certainly  associated.  This  does  not  mean 
that  the  value  of  the  methylene  group  is  therefore  certainly  in 
error  because  associated  compounds  can  give  this  as  well  as  others, 
provided  the  homologues  are  equally  associated;  and  even  if  they 
are  unequally  associated,  the  average  value  for  the  methylene 
grouping  may  not  be  greatly  in  error  although  the  individual 
differences  may  be  large.  Finally  the  fact  that  the  calculated 
values  in  Table  XXV  differ  from  the  observed  values  by  less  than 
1  per  cent  seems  to  put  a  maximum  limit  upon  certain  kinds  of 
constitutive  influences. 

Hitherto  it  has  been  deemed  necessary  to  give  oxygen  a  differ- 


124  FLUIDITY  AND  PLASTICITY 

ent  value  depending  upon  whether  the  oxygen  was  in  a  carbonyl 
group,  hydroxyl,  ether,  et  cetera.  We  will  now  attempt  to  show 
that  this  was  necessary  so  long  as  viscosities  formed  the  basis  of 
comparison,  but  it  was  not  an  evidence  of  constitutive  influence, 
and  in  comparing  fluidities  only  one  value  for  oxygen  is  obtained 
irrespective  of  the  manner  in  which  it  is  combined,  and  yet  we 
have  seen  that  satisfactory  association  factors  are  obtained. 
Let  A B  and  A'B'  in  Fig.  47  represent  two  fluidity  curves,  parallel 
to  each  other  and  therefore  presumably  representing  members 
of  the  same  class  of  substances,  and  let  a  third  fluidity  curve 
CD  be  at  an  angle  to  the  other  two  to  represent  a  substance  in 
another  class.  Since  we  have  elected  to  compare  absolute 
temperatures  at  a  fluidity  of  200,  this 
amounts  to  comparing  the  intercepts 
of  the  curves  on  the  line  AD,  whose 
equation  is  <p  =  200.  The  corre- 
sponding viscosity  curves  obtained 
by  taking  the  reciprocal  values  of 
the  above  fluidities  and  multiplying 

0  — -      by  10,000  are  represented  by  ab, 

a'b'   and  cd\  and  ad  is  of  course  the 

FIG.  47. — Diagram  illustrating  /  ' 

the  relationship  between  viscos-    reciprocal  of  the  line  AD.     But  the 

ity- and  fluidity-temperature  point  a  and  the  point  wnere  the 
curves.  . 

curve  a'b'  crosses  ad  are  points  of 

equal  slope  on  the  viscosity  curves,  hence  within  a  given  class 
it  makes  no  great  difference  whether  we  compare  temperatures 
corresponding  to  a  given  fluidity  or  temperatures  corresponding 
to  a  given  slope  on  the  viscosity  curves.  The  latter  is  exactly 
the  method  of  comparison  which  Thorpe  and  Rodger  found 
very  advantageous.  But  between  different  classes  they  found 
difficulties  which  they  attributed  to  constitutive  influences.  But 
their  difficulty  is  now  easily  explained,  for  d  is  the  true  reciprocal 
of  the  point  D  which  we  believe  should  be  used  in  the  com- 
parison; on  the  other  hand,  they  selected  the  point  e,  which  has 
the  same  slope  as  the  point  a,  and  for  this  choice  we  think  that 
there  is  not  adequate  reason. 

It  has  been  customary  to  assume  that  different  kinds  of 
groupings  should  have  special  values  assigned  to  them,  and 
particularly  important  among  these  was  the  "ring  grouping." 


FLUIDITY  AND  THE  CHEMICAL  COMPOSITION         125 

But  it  is  not  clear  that  this  is  unavoidable,  for  in  the  case  of 
benzene  we  note  that  the  compound  differs  from  hexane  by  eight 
hydrogen  atoms,  and  since  we  found  in  the  allyl  compounds  that 
the  absence  of  a  pair  of  hydrogen  atoms  is  compensated  for  to  the 
extent  of  114.4  we  obtain  for  the  calculated  value,  59.2  X  6  — 
95.7  X  6  +  114.4  X  4  =  238.6.  The  observed  value  is  311.9; 
hence  the  association  is  1.30,  which  is  somewhat  larger  than  the 
value  obtained  by  Traube  of  1.18.  It  has  usually  been  believed 
that  the  more  compact  and  symmetrical  the  molecule  was,  the 
lower  would  be  the  temperature  required  to  give  it  a  certain 
fluidity.  In  disregarding  constitutive  influences  entirely  for  the 
time  being,  as  we  have  done  here,  we  suppose  that  benzene  would 
require  the  same  temperature  as  a  straight  chain  hydrocarbon 
containing  four  "double  bonds."  If  Traube's  value  of  the 
association  or  some  other  value  less  than  1.30  is  correct,  we  will 
be  compelled  to  assign  a  positive  value  to  the  ring  grouping  in 
order  to  increase  the  calculated  value.  In  other  words,  the 
evidence  at  hand  indicates  that  the  effect  of  the  ring  grouping 
is  not  to  make  the  compound  less  viscous  but  more  so.  This  is 
so  contrary  to  earlier  belief  and  to  the  probabilities  of  the  case 
that  it  seems  preferable  to  await  further  data  before  assigning 
any  value  to  the  ring  grouping. 

Having  been  unable  to  detect  the  effect  of  constitutive 
influences  upon  fluidity  with  the  data  at  hand  in  the  halogen, 
oxygen,  or  ring  compounds,  we  have  left  remaining  one  positive 
evidence  in  the  value  which  we  have  found  it  necessary  to  give 
to  the  "iso-grouping."  This  effect  is  not  large  but  it  is  fairly 
uniform  and  quite  outside  of  the  observational  error.  We  cannot 
believe  that  normal  hexane  and  heptane  are  sufficiently  associated 
to  account  for  the  higher  temperature  above  their  isomers 
required  to  give  them  a  fluidity  of  200.  If  then  an  iso-grouping 
affects  fluidity  it  is  probable  that  there  are  other  constitutive 
influences,  but  the  solution  of  this  problem  evidently  requires 
more  data,  particularly  among  the  higher  homologues.  In  this 
connection  the  reader  should  have  regard  for  the  relation  of 
fluidity  to  volume,  to  be  discussed  later. 

Before  closing  the  chapter  on  fluidity  and  chemical  composition 
and  constitution,  we  may  add  that  constants  calculated  for  a 
fluidity  of  300  give  an  association  which  is  invariably  a  little 


126 


FLUIDITY  AND  PLASTICITY 


lower  than  at  a  fluidity  of  200.  This  is  as  one  would  expect. 
From  these  values  of  the  association  the  temperature  coefficient 
of  the  association  factor  can  be  obtained.  (Bingham  (1910), 
page  306.) 

From  the  constants  at  different  fluidities  one  may  conceivably 
obtain  a  series  of  points  on  the  fluidity-temperature  curve 
heretofore  unknown  perhaps,  and  using  these  points  the  whole 
curve  may  evidently  be  drawn. 

The  constants  for  fluidities  200  and  300  are  for  convenience 
grouped  together  in  Table  XXXIII. 

TABLE  XXXIII. — TEMPERATURE  CONSTANTS  AT  FLUIDITY  200  AND 
FLUIDITY    300 


Atom  or  grouping 

v  =  200 

v  =  300 

Carbon 

—95  7 

-110  2 

Hydrogen  
Oxygen  
Iso  ... 

59.2 
24.2 
-7.6 

67.8 
27.1 

-8.2 

Double  bond  
Sulfur  
Chlorine  

114.4 
76.5 
109.5 

131.3 

CHAPTER  III 

FLUIDITY    AND  TEMPERATURE,  VOLUME  AND  PRES- 
SURE ;  COLLISIONAL  AND  DIFFUSIONAL  VISCOSITY 

We  have  established  in  the  preceding  chapter  that  the  vis- 
cosity of  a  substance  is  closely  dependent  upon  the  magnitude 
of  the  molecules  making  up  the  substance.  In  this  and  succeed- 
ing chapters  we  will  investigate  the  relation  between  viscosity 
and  various  physical  properties. 

Temperature. — Prior  to  1800,  water  was  considered  to  be 
perfectly  fluid,  but  by  causing  equal  volumes  of  water  at  corre- 
sponding pressures  to  flow  through  tubes  of  given  dimensions 
Gerstner  in  that  year  proved  that  the  fluidity  (Fliissigkeit) 
of  water  varies  considerably  with  the  temperature. 

We  have  already  seen  that  Poiseuille  expressed  this  change  in 
the  form  of  the  parabolic  equation 

K  =  a  +  bTc  +  cTV.  (36) 

After  viscosity  had  been  denned,  O.  E.  Meyer  (1861)  introduced 
the  viscosity  coefficient  into  the  formula  which  then  became 


(37) 


where  T\O  is  the  viscosity  at  0°C  and  Tc  is  the  temperature 
Centigrade.  In  spite  of  the  fact  that  the  two  equations  are  not 
interchangeable,  the  latter  formula  is  usually  associated  with  the 
name  of  Poiseuille.  We  will  refer  to  it  as  tne  Meyer-Poiseuille 
formula.  It  holds  for  water  from  0  to  45°  with  a  maximum 
deviation  of  1  per  cent.  For  temperatures  above  45°  Meyer  and 
Rosencranz  (1877)  proposed  the  formula 


Various  investigators  have  employed  the  Meyer-Poiseuille 
formula  and  confirmed  the  fact  of  its  limited  applicability.  We 
may  mention  Grotrian  (1877),  Noack  (1886),  Thorpe  and 
Rodger  (1893),  Knibbs  (1895). 

127 


128  FLUIDITY  AND  PLASTICITY 

In  1881  Slotte  gave  a  formula  to  cover  the  entire  range  of 
viscosities  from  0  to  100° 

a  -  bTc 


~  c  +  Tc 
or 


(39) 


which  accords  with  the  values  of  Sprung  to  0.7  per  cent. 

Most  of  the  formulas  which  have  been  proposed  have  been 
applied  primarily  to  water.  But  Koch  (1881)  and  Wagner  (1883) 
found  that  a  formula  of  a  different  type  is  necessary  for  mercury 
and  Koch  proposed  the  formula 

i,  =  a  +  $TC  +  77V  +  dTc*  (41) 

which  holds  from  —20  to  340°C  with  a  maximum  deviation  of 
less  than  2  per  cent.  But  Slotte  has  applied  the  much  simpler 
formula  of  Meyer  and  Rosencranz  with  good  results.  On  the 
other  hand,  Batschinski  (1900)  has  given  a  formula  for  mercury 

r,  =  aT  +  6  +  cT  (42) 

where  T  is  the  temperature  absolute.     As  a  first  approximation 

•nT  =  a,  (43) 

which  can  be  deduced  from  Jaeger's  theory  of  fluid  friction. 

Graetz  (1883-5)  is  one  of  the  few  who  have  attempted  to  derive 
a  formula  from  theoretical  considerations.  We  may  therefore 
give  his  argument  in  some  detail. 

According  to  Maxwell  (1868)  the  viscosity  of  a  body  is  the 
product  of  two  factors,  the  modulus  of  rigidity  E  and  the  time 
of  relaxation  T.  The  time  of  relaxation  was  defined  as  the  time 
necessary  for  the  strain  after  deformation  in  a  body  to  sink  to 
l/E  of  its  original  value.  The  reciprocal  of  the  time  of  relaxation 
is  called  the  relaxation  number,  n,  or 


This  is  the  number  of  times  per  second  that  the  strain  will  sink 
to  l/E  per  second  if  the  strain  is  renewed.  For  absolutely  rigid 
solids  the  value  of  T  is  infinite  and  for  ductile  solid  bodies  which 
show  elastic  after-effect  the  relaxation  may  continue  for  hours  or 
days.  But  if,  through  raising  the  temperature,  the  substance  is 


FLUIDITY  AND  TEMPERATURE  129 

changed  to  a  liquid  or  gas,  the  time  of  relaxation  becomes  smaller 
and  smaller,  and  for  air  Maxwell  has  given  the  value 

T  =  1/5,099,100,000  sec. 

With  rising  temperature,  the  value  of  T  decreases,  and  according 
to  Graetz,  one  may  write 


where  $  is  the  temperature  reckoned  from  the  temperature  at 
which  the  viscosity  is  infinite,  i.e.,  the  temperature  of  solidifica- 
tion. 

In  gases  the  modulus  of  rigidity  is  known  to  be  equal  to  the 
gas  pressure  at  the  critical  temperature  and  is  of  the  order  of 
magnitude  of  a  hundred  atmospheres.  In  solids,  where  the 
modulus  of  rigidity  is  known,  it  has  a  value  from  100,000  to 
1,000,000  atmospheres.  Since  E  decreases  in  passing  from  the 
solid  through  the  liquid  into  the  gaseous  condition,  its  value 
approaches  the  critical  pressure  P  at  the  critical  temperature  #0, 
and  we  have  according  to  Graetz 

E    =   P   +   &!(#„    -   #)    +   &2(#0    -    #)2   +   &3(#0    -   #)3   +  •      •      - 

where  &i,  &2,  63,   .    .    .   are  constants. 
From  Maxwell,  we  get 

,  =  Er  =  - 
n 

or 

P  +  &!(#  o  -  t?)  +  .    .    . 


where  a,  0i,  /32,  .  .  .  are  constants.  Since  the  formula  with  a 
large  number  of  constants  is  of  little  practical  use,  Graetz  neg- 
lected the  constants  0i,  j82,  .  .  .  which  are  of  small  magnitude 
and  thus  obtained 


or  if  the  temperatures  are  changed  to  the  absolute  scale, 

T7     —  T7 


(44) 


130  FLUIDITY  AND  PLASTICITY 

where  Tcr  is  the  critical  temperature  and  Ts  is  the  temperature  of 
solidification. 

Applying  this  formula  to  the  data  of  Rellstab,  and  Pribram 
and  Handl,  Graetz  found  that  it  was  satisfactory  in  some  fifty 
cases,  but  it  is  inapplicable  to  the  fatty  alcohols. 

This  formula  is  a  particular  form  of  the  one  already  given  by 
Slotte.  Slotte  (1883-90)  reached  the  conclusion  that  none  of 
the  preceding  formulas  gives  satisfactory  results  with  substances 
whose  viscosity  changes  rapidly  with  the  temperature,  as  is 
generally  true  of  very  viscous  substances.  He  proposed  the 
formula 


(45) 


where  T  is  the  temperature  centigrade  and  a,  b,  and  n  are  arbi- 
trary constants.  Slotte  found  that  this  formula  gave  better 
results  than  any  other  and  Thorpe  and  Rodger  adopted  it  in 
their  great  work  as  the  most  satisfactory  formula  at  their  disposal, 
but  this  formula  like  the  others  breaks  down  when  applied  to  the 
alcohols.  In  the  case  of  several  of  the  alcohols  it  was  necessary 
to  apply  the  formula  three  times  with  different  constants  over 
different  parts  of  the  curves  in  order  to  reproduce  the  observed 
values  with  anything  like  the  desired  accuracy.  The  values  of 
n  vary  from  1.4  to  4.3. 

Several  exponential  formulas  have  been  proposed.     Reynolds 
(1886)  and  Stoel  (1891)  suggested  the  formula 

,,  =  ae-yTc  (46) 

which  Reynolds  found  to  apply  to  olive  oil  and  Stoel  and  De  Haas 
(1894)  found  to  apply  to  methyl  chloride  between  the  boiling- 
point  and  the  critical  temperature.  In  this  formula  e  is  the 
natural  logarithmic  base  and  a  and  y  are  constants.  Heydweiller 
in  1895  investigated  benzene  and  ethyl  ether  over  a  similar 
range  of  temperature,  and  found  that  the  formula  holds  between 
the  reduced  temperatures  of  0.62  and  0.87  for  the  purposes  of 
interpolation  ;  but  that  similar  systematic  deviations  occur  for  all 
substances,  which  is  due  to  the  fact  that  the  temperature  coef- 
ficient of  the  viscosity  is  not  constant  but  passes  through  a  point  of 
inflection.  Below  the  boiling-point  the  viscosity-temperature 
curve  is  convex  toward  the  temperature  axis  while  near  the 


FLUIDITY  AND  TEMPERATURE  131 

critical  temperature  it  is  concave,  and  the  deviations  from  the 
formula  are  considerable.  Heydweiller  made  the  interesting 
observation  that  within  the  series  of  compounds  with  which  he 
worked  the  temperatures  of  equal  viscosity  are  in  the  ratio  of 
their  critical  temperatures.  But  to  this  rule  water  and  the 
alcohols  are  exceptional. 

De  Heen  in  his  Theorie  des  Liquides  (1888)  found  the  following 
formula  satisfactory 

77  =  T7o(l  +  ae~bTcY  (47) 

The  constant  c  varies  little  from  liquid  to  liquid  from  2.65  to  2.85. 

Jaeger  (1893)  has  worked  out  a  kinetic  theory  of  liquids  on 
the  ground  that  the  transfer  of  momentum  takes  place  by  the 
molecules  passing  back  and  forth  from  one  layer  of  molecules  to 
another.  He  gives  the  expression 


where  r  is  the  radius  of  a  molecule,  v  its  mean  velocity,  X  its 
mean  free  path,  and  p  is  the  density  of  the  liquid. 

Similarly  Kamerling  Onnes  has  derived  a  formula  from  the 
theory  of  corresponding  condition  of  van  der  Waals, 

=  Constant  (49) 


where  V  is  the  molecular  volume,  M  the  molecular  weight, 
Tcr  the  critical  temperature,  and  r/F%  is  the  molecular  surface 
friction.  The  formula  does  not  apply  at  low  temperatures  and 
perhaps  only  perfectly  as  the  critical  temperature  is  reached. 

Perry  (1893)  states  that  sperm  oil  cannot  be  represented  by 
any  single  formula  since  a  discontinuity  occurs  in  the  viscosity 
and  density  curves  at  40°.  It  should  be  added  that  he  took  care 
that  the  velocity  of  flow  did  not  exceed  the  critical  value  for 
viscous  flow.  He  employed  two  sets  of  constants  in  the  formula 
77  =  a(T  -  6)~c  (50) 

Examining  a  considerable  number  of  the  formulas  which 
had  already  proved  of  value  and  given  above,  Duff  (1896) 
obtained  the  following  formula  by  integrating  the  curve  of 
subtangents  derived  from  them: 


132 


FLUIDITY  AND  PLASTICITY 


This  formula  reproduces  observed  values  of  all  classes  of  com- 
pounds with  remarkable  fidelity,  but  it  contains  four  constants. 
Finally  Batschinski  (1901)  has  indicated  that  the  viscosity 
varies  inversely  as  the  cube  of  the  absolute  temperature 

*  =  |3  (52) 

where  E  is  dependent  upon  the  nature  of  the  liquid  and  is  called 
"the  viscosity  parameter."  Using  the  data  of  Thorpe  and 
Rodger,  Pribram  and  Handl,  Gartenmeister  and  others,  Bats- 
chinski has  tested  his  expression  fully.  While  it  gives  very  good 


7 


Temperature  absolute 
FIG.  48. — Diagram  illustrating  ideal  fluidity-temperature  curves. 

agreement  in  many  cases,  there  are  numerous  exceptions,  particu- 
larly water,  the  alcohols,  acids  and  anhydrides  (c/.  Eq.  (41) 
for  mercury). 

Fluidity  and  Temperature. — From  the  evidence  given  in  Chap- 
ter I  to  prove  that  fluidities  are  normally  additive  in  homogeneous 
mixtures,  it  would  appear  probable  that  fluidity-temperature 
curves  in  their  simplest  form  are  straight  lines,  and  since  every 
liquid  is  theoretically  capable  of  existing  in  an  undercooled 
condition,  these  curves  meet  at  absolute  zero.  The  equation  for 
the  fluidity-temperature  curves  of  all  substances  should  be 

v=  aT 

which  is  the  same  as  Eq.  (43).  A  family  of  curves  would  present 
the  appearance  shown  diagrammatically  in  Fig.  48. 


FLUIDITY  AND  TEMPERATURE 


133 


Unfortunately,  fluidity-temperature  curves  are  not  generally 
linear  throughout.  Even  the  aliphatic  hydrocarbons,  which  are 
supposedly  unassociated,  give  curves  which  depart  considerably 
from  linearity.  For  this  peculiarity  two  possible  explanations 
suggest  themselves.  The  effect  of  expansion  is  not  linear  except 
in  the  case  of  mercury  and  there  may  be  changes  in  the  molecular 
weight,  either  association  or  dissociation.  Later  each  of  these 
causes  will  be  considered  in  detail.  Suffice  it  for  the  present 
to  note  that  the  alcohols,  Fig.  46,  give  fluidity-temperature 
curves  which  are  strongly  curved  at  low  fluidities,  but  the  curves 
tend  to  become  linear  at  high  fluidities  as  is  true  of  the  other 

classes  of  compounds.     In    a        . 

given  homologous  series  the  evi- 
dence at  hand  shows  a  tendency 
for  the  fluidity-temperature 
curves  near  the  ordinary  boil- 
ing-points to  become  parallel 
to  each  other  with  equal  dis- 
tances between  them.  But 
they  must  all  meet  at  absolute 


zero    of    temperature,    which 
requires   that    in    the    higher 


Temperature  absolute 
FIG.    49.— Diagram    illustrating    the 
fluidity-temperature  curves  of  a  "  homol- 
ogous series"  of  liquids. 


members  of  the  series  at  least 
there  should  be  a  region  of 
rapid  curvature.  This  is  the 
region  of  " softening"  which  has  been  observed  in  many  viscous 
substances.  According  to  the  views  here  presented  the  phe- 
nomenon of  softening  may  not  occur  in  any  substance  con- 
sisting of  monatomic  molecules  but  will  certainly  become 
manifest  as  we  increase  the  complexity  of  the  molecule.  These 
ideas  are  represented  diagrammatically  in  Fig.  49.  If  they 
represent  accurately  the  actual  behavior  of  substances,  the 
fluidity-temperature  curves  have  the  following  properties:  (1) 
They  all  approach  linearity  as  the  fluidity  increases,  i.e.,  each 
tends  to  become  asymptotic  to  a  line  which  forms  an  acute 
angle  with  the  temperature-axis.  (2)  As  the  fluidity  becomes 
very  small,  each  curve  tends  to  become  asymptotic  to  the  tem- 
perature-axis itself.  Thus  the  general  form  of  the  fluidity- 
temperature  curve  must  be  a  hyperbola,  since  this  is  the 


134  FLUIDITY  AND  PLASTICITY 

curve  which  contains  these  properties.  The  linear  curve  for 
substances  like  ^mercury  is  of  course  but  a  special  and  extreme 
case.  It  is  very  important  that  we  find  out  whether  other  mon- 
atomic  liquids,  such  as,  for  example,  liquid  sodium,  argon,  et  cet. 
have  a  linear  volume  temperature  and  fluidity-temperature  curve. 
Given  the  properties  of  a  general  fluidity-temperature  curve 
it  is  easy  to  obtain  the  curve  which  contains  these  properties. 
The  simplest  is 

T  =  A<p  +  C  -  |  (53) 

For  the  simplest  case,  which  we  have  in  mercury,  the  constants 
B  and  C  are  each  zero  and  our  equation  becomes  that  of  Batschin- 
ski  (42);  for  other  substances  at  high  temperatures  the  term  B/<p 
becomes  negligible  and  the  equation  becomes  that  of  the  asymp- 
tote 

T  =  A<p  +  C 

This  linear  equation  can  be  transformed  into  the  formula  of 
Meyer  and  Rosencranz.  Thus  while  our  formula  (53)  agrees 
with  some  of  the  simpler  formulas  which  have  been  proposed, 
it  is  at  variance  with  those  which  have  had  the  widest  usefulness. 
For  example,  the  Meyer-Poiseuille  formula  may  be  written 

<p  =  A  +  BTC  +  C7V  (53a) 

which  is  the  equation  of  a  parabola.  Were  it  valid  as  a  general 
equation,  the  fluidity  would  have  a  minimum  value  at  0°C, 
and  at  high  temperatures  the  fluidity  curve  would  tend  to  be- 
come parallel  to  the  fluidity-axis.  It  needs  no  argument  to  show 
that  this  equation  cannot  be  general. 

By  increasing  the  number  of  terms,  the  formula  of  Meyer- 
Poiseuille  becomes  the  second  formula  of  Slotte, 

v  =  A(B  +  Tc)n 

The  effect  of  the  additional  terms  is  to  increase  the  curvature 
but  it  does  not  remedy  the  defects  noted.  The  equation  of 
Stoel  and  others  may  be  regarded  as  a -special  case  of  Slotte's 
equation  where  the  number  of  terms  is  infinite. 

The  formula  of  Graetz  is  defective  in  that  it  assumes  that  the 
fluidity  becomes  zero  at  a  definite  positive  temperature  and  that 
at  the  critical  temperature  the  fluidity  is  infinite.  Neither  of 
these  assumptions  can  longer  be  held. 


FLUIDITY  AND  TEMPERATURE 


135 


TABLE  XXXIV. — THE  CONSTANTS  IN  THE  EQUATION  T  =  A<p  +  C  - 


Substance 

A 

B 

C 

Mean    per 
cent    differ- 
ence 

1 

Water. 

0.27727 

1,263.3 

278.80 

0.17 

2 

Bromine  

0.79098 

1,376.3 

227.16 

0.03 

3 

Nitrogen  peroxide 

0.32274 

3  ,  837  .  6 

231.92 

0.00 

4 

Pentane 

0.16544 

14  ,  937  .  0 

256  .  61 

0.03 

5 

Isopentane 

0  .  18327 

10,454.0 

234  .  17 

0.03 

6 

Hexane  

0.21892 

9,137.8 

254.03 

0.04 

7 

Isohexane  

0.20708 

9,562.8 

252.85 

0.05 

8 

Heptane  

0.23203 

8,681.2 

272.70 

0.09 

9 

Isoheptane  

0.22445 

869.84 

267.35 

0.12 

10 

Octane  

0.27055 

6,332.6 

277.25 

0.16 

11 

Trimethyl  ethylene  

0.21942 

6,770.3 

203.56 

0.06 

12 

Isoprene  

0.21505 

7,126.5 

208.74 

0.09 

13 

Diallyl  

0.19818 

9,774.4 

247.75 

0.06 

14 

Methyl  iodide  

0.44916 

3,107.1 

215.85 

0.01 

15 

Ethyl  iodide  

0.43828 

4,348.2 

243.29 

0.04 

16 

Propyl  iodide  

0.46571 

3,474.4 

255.85 

0.11 

17 

Isopropyl  iodide  

0.41279 

4,108.8 

262  .  07 

0.14 

18 

Isobutyl  iodide  

0.46881 

2,727.2 

264.30 

0.23 

19 

Allyl  iodide  

0.47443 

2,994.0 

249.80 

0.11 

20 

Ethyl  bromide  

0.35853 

4,073.2 

217.39 

0.02 

21 

Propyl  bromide  

0.36084 

4,950.6 

248.95 

0.04 

22 

Isopropyl  bromide  

0.33069 

5,009.9 

248.57 

0.03 

23 

Isobutyl  bromide  

0.34098 

4,754.4 

270.75 

0.20 

24 

Allyl  bromide 

0.36448 

4,467.7 

241.77 

0.06 

25 

Ethylene  bromide  

0.68897 

1,421.6 

277.92 

0.23 

26 

Propylene  bromide  

0.64567 

1,463.0 

278.53 

0.34 

27 

Isobutylene  bromide  

0.65358 

1,060.2 

288.40 

0.04 

28 

Acetylene  bromide  

0.63374 

2,275.3 

249.46 

0.12 

29 

Propyl  chloride  

0.25540 

7,465.2 

246.74 

0.05 

30 

Isopropyl  chloride  

0.24993 

5,881.9 

234.22 

0.01 

31 

Isobutyl  chloride  

0.28656 

4,973.0 

253.00 

0.02 

32 

Allyl  chloride  

0.26292 

6,377.2 

234.10 

0.03 

33 

Methylene  chloride  

0.39806 

2,666.5 

212.97 

0.04 

34 

Ethylene  chloride  

0.44121 

2,219.3 

258.83 

0.08 

35 

Ethylidene  chloride  

0.33277 

4  ,  651  .  6 

247  .  99 

0.04 

36 

Chloroform  

0.40697 

4,400.0 

245  .  73 

0.07 

37 

Carbon  tetrachloride  

0.47337 

1,807.5 

262.15 

0.08 

38 

Carbon  dichloride  

0.55768 

3,081.6 

259.13 

0.20 

39 

Carbon  disulfide  

0  .  26901 

16,751.0 

282.23 

0.17 

40 

Methylsulfide  

0.25514 

8,387.6 

230.57 

0.01 

41 

Ethylsulfide  

0.28517 

6,447.9 

258.00 

0.11 

42 

Thiophene  

0.38204 

2,967.2 

254.95 

0.09 

43 

Dimethylketone  

0.23871 

8,905.0 

247.64 

o.oa-' 

44 

Methylethylketone  

0.27275 

5,572.6 

252.32 

0.08 

45 

Diethylketone  

0.28145 

6,179.3 

262.41 

0.16 

46 

Methylpropylketone  

0.29251 

5,544.3 

263.21 

0.15 

47 

Acetaldehyde 

0.15205 

18,364.0 

265.13 

0.006 

48 

0.52465 

963.4 

281.57 

0.19 

136 


FLUIDITY  AND  PLASTICITY 
TABLE  XXXIV.— (Continued) 


Substance 

A 

B 

C 

Mean    per 
cent    differ- 
ence 

49 

Acetic  acid  

0.42437 

2,716.8 

291.81 

0.22 

50 

Propionic  acid  

0.43533 

2,908.9 

287  .  53 

0.41 

51 

Butyric  acid  

0.45359 

1,951.9 

296.43 

0.69 

52 

Isobutyric  acid  

0.45841 

2,143.6 

288.41 

0.43 

53 

Acetic  acid  anhydride  

0.40620 

2,747.8 

273.61 

0.31 

54 

Propionic  acid  anhydride  

0.39705 

2,444.6 

287.45 

0.61 

55 

Diethy  ether  

0.  16574 

14,674.0 

256.72 

0.07 

56 

Benzene  

0.32052 

2,633.1 

260.82 

0.05 

57 

Toluene  

0.32688 

4,193.5 

262.66 

0.12 

58 

Ethyl  benzene  

0.34180 

4,540.8 

273.54 

59 

Ortho  xylene  

0.37738 

3,009.3 

271.96 

0.28 

60 

Meta  xylene  

0.34134 

4,542.9 

266.82 

0.19 

61 

Para  xylene  

0.32087 

5,127.4 

277.17 

0.20 

62 

Methyl  alcohol  

0.24316 

4,498.9 

279  .  01 

0.14 

63 

Ethyl  alcohol  

0.28395 

2,398.6 

298  .  39 

0.12 

64 

Propyl  alcohol  

0.31496 

1,211.7 

308.41 

0.35 

65 

Isopropyl  alcohol  

0.29810 

738.16 

300.17 

0.36 

66 

Butyl  alcohol  

0.33610 

877.08 

311.94 

0.72 

67 

Isobutyl  alcohol  

0.33648 

512.18 

309.72 

0.81 

68 

Trimethyl  carbinol  

0.29657 

260.86 

305.73 

0.33 

69 

Amyl  alcohol,  active  

0.35020 

376.48 

311.50 

1.17 

70 

Amyl  alcohol,  inactive  

0.36060 

513.18 

312.40 

1.08 

71 

Dimethyl  ethyl  carbinol  

0.29578 

259.87 

307  .  20 

0.87 

72 

Allyl  alcohol 

0.28815 

1,935.7 

299  .  53 

0.23 

73 

Methyl  formate 

0.34444 

1  ,  292  .  0 

198.31 

0.01 

74 

Ethyl  formate  

0.29418 

4,858.1 

239  .  32 

0.02 

75 

Propyl  formate  

0.29797 

4,800.6 

260  .  44 

0.06 

76 

Methyl  acetate  

0.26047 

6,475.7 

249.48 

0.03 

77 

Ethyl  acetate  

0.27056 

5,361.2 

257.20 

0.06 

78 

Propyl  acetate  

0.29534 

4,262.6 

267.50 

0.15 

79 

Methyl  propionate  

0.27300 

5,954.3 

261.08 

0.15 

80 

Ethyl  propionate  

0.29125 

4,846.4 

264.51 

0.12 

81 

Methyl  butyrate  

0.30210 

4,315.3 

265.89 

0.15 

82 

Methyl  isobutyrate  

0.28615 

5,073.2 

264.44 

0.15 

83 

Methyl  propyl  ether  

0.21797 

7,206.3 

224.27 

0.03 

84 

Ethyl  propyl  ether  

0.20872 

8,933.6 

255.80 

0.05 

85 

Dipropyl  ether  

0.23579 

6,858.5 

266.34 

0.14 

86 

Methyl  isobutyl  ether  

0.21201 

8,748.8 

250.76 

0.03 

87 

Ethyl  isobuty  ether  

0.22545 

7,188.2 

260.96 

0.09 

We  will  now  see  how  far  the  fluidity  Eq.  (53)  can  be  used  to 
reproduce  the  experimentally  observed  values.  In  Table 
XXXIV  we  give  the  constants  for  the  87  substances  investigated 
by  Thorpe  and  Rodger  and  in  the  last  column  of  the  table  we 
give  the  average  percentage  difference  between  the  observed 
and  calculated  values. 

The  mean  percentage  difference  between  the  calculated  and 


FLUIDITY  AND  TEMPERATURE 


137 


observed  values  is  0.17  for  the  87  substances  and  based  on  some 
1,000  duplicate  observations.  If  we  omit  the  alcohols,  this 
difference  falls  to  0.09  for  70  substances.  This  is  much  better 
agreement  than  Thorpe  and  Rodger  obtained  with  Slotte's 
equation,  since  the  percentage  difference  is  nearly  twice  the 
above,  viz.,  0.15  per  cent  for  64  substances.  But  the  real  test 
is  with  substances  which  give  fluidity  curves  departing  widely 
from  the  linear  type  and  here  Slotte's  equation  breaks  down 
completely. 

For  this  type  of  substances,  the  fluidity  Eq.  (53)  with  three 
constants  does  not  reproduce  the  observed  values  to  the  limit  of 
experimental  error,  but  a  great  improvement  can  be  made  by 
introducing  another  constant  and  writing  the  equation 


For  example,  the  mean  divergence  between  the  observed  and 
calculated  values  for  the  eight  substances,  which  gave  the  largest 
percentage  difference,  was  0.77  per  cent  with  the  simpler  for- 
mula; the  Eq.  (54)  with  four  constants  reduces  this  to  only 
0.07  per  cent  which  is  nearly  within  the  limits  of  the  experi- 
mental error.  In  the  case  of  water,  which  gave  a  mean  difference 
of  only  0.17  per  cent  with  the  simpler  formula,  the  difference  is 
reduced  to  0.01  per  cent  and  similarly  in  the  case  of  octane  it  is 
reduced  from  0.16  to  0.02  per  cent.  For  reference,  the  constants 
for  these  substances  are  given  in  Table  XXXV. 


TABLE  XXXV.— THE  CONSTANTS  IN  THE  EQUATION  T  =  A<?  +  C  - 


Mean 

Substance 

A 

B 

C 

D 

per  cent 

difference 

Water  !   0  .  23275 

8,676.8 

309.17 

120 

0.01 

Octane  

0.14507 

100,745.0 

438.10 

400 

0.02 

Butyric  acid  

0.16154 

70,630.0 

504.44 

250 

0.02 

Isobutyric  acid  

0.23862 

43,665.0 

433.17 

200 

0.06 

Propionic  acid  anhydride  

0.23619 

52,294.0 

425.82 

250 

0.07 

Butyl  alcohol  

0.23695 

4,802.0 

349.71 

40 

0.04 

Isobutyl  alcohol  

0.23700 

2,993.7 

340.66 

30 

0.09 

Active  amyl  alcohol  

0.24650 

2,942.8 

346.82 

30 

0.08 

Inactive  amyl  alcohol  

0.24191 

3,908.7 

354  .  17 

35 

0.09 

Dimethyl  ethyl  carbinol  

0.22988 

2,124.0 

328.84 

30 

0.09 

138  FLUIDITY  AND  PLASTICITY 

Fluidity  and  Pressure. — To  find  the  effect  of  pressure  on 
viscosity,  Coulomb  in  1800  measured  the  rate  of  oscillation 
of  a  disk  in  a  liquid  both  under  atmospheric  pressure  and  when 
the  space  above  the  liquid  had  been  evacuated.  It  was  found 
that  the  viscosity  is  independent  of  small  changes  of  pressure. 
This  conclusion  was  confirmed  by  Poiseuille  in  1846,  using  his 
transpiration  method. 

However,  quite  the  opposite  conclusions  must  be  drawn 
from  the  experiments  of  Warburg  and  Babo  (1882),  but  they 
employed  liquid  carbon  dioxide  at  25.1°C,  which  is  quite  near  the 
critical  temperature,  and  they  used  pressures  from  70  to  105 
atmospheres.  It  is  worth  noting  that  under  these  conditions 
the  compressibility  of  carbon  dioxide  is  0.00314  which  is  about 
18  times  as  great  as  that  of  ether  at  the  same  temperature.  They 
found  an  increase  in  the  viscosity  which  amounted  to  over  25 
per  cent,  and  it  therefore  seemed  possible  that  the  effect  was 
caused  by  the  change  in  density  and  that  a  similar  effect  would  be 
observed  in  other  liquids  if  high  enough  pressures  were  employed. 

Warburg  and  Sachs  (1884)  continued  the  previous  investigation 
and  indeed  found  that  liquid  carbon  dioxide,  ether,  and  benzene 
all  suffer  an  increase  in  viscosity  on  increasing  the  pressure,  but 
they  also  noted  that  water  is  exceptional  in  that  an  increase  in 
pressure  actually  lowers  the  viscosity.  They  sought  to  connect 
the  viscosity  and  pressure  by  means  of  the  following  linear 
formula, 

7?     =    T,o(l    +    AP).  (55) 

The  values  of  the  constant  A  are  given  in  Table  XXXVI. 
TABLE  XXXVI. — CONSTANTS  IN  EQUATION  (53) 


Substance 

Carbon 
dioxide 

Ether 

Benzene 

Water 

Temperature  of  experiment  
Critical  temperature  
A  X  106 

25.1 
30.9 
7  470 

20 
190 
730 

20 
280.6 
930 

20 
365 
—  170 

The  striking  fact  that  water  is  peculiar  in  this  as  in  so  many 
other  respects  was  discovered  independently  by  Rontgen  (1884). 
It  was  made  the  subject  of  a  special  study  by  Cohen  in  1892, 


FLUIDITY  AND  TEMPERATURE 


139 


working  at  1,  5,  and  23°  at  pressures  ranging  from  1  to  600 
atmospheres  and  using  pure  water  and  four  solutions  of  sodium 
chloride  of  different  concentrations.  The  nature  of  his  results 
is  shown  in  Figs.  50  to  54.  In  Fig.  50  the  percentage  change 
in  the  time  of  flow,  [(ti  —  <p)/£i]100,  is  plotted  as  ordinates 

FIG.  53. 


FIG.  50. 


^S 

X 

<?3C 

- 

\ 

K      ^ 

"*v. 

^° 

^ 

\ 

\ 

\,o 

3    ?00   400    600    800  WOO 

\o°\^*' 

^  

.  —  x 
^,x 

^ 

^ 

"  \&*' 

^ 
/ 

/* 

/ 

'& 

/ 

/ 

/ 

/ 

A         t 

8      12      16     ZQ     24 
FIG.  51. 

FIG.  52. 


FIG.  54. 


The  effect  of  pressure  on  the  viscosity  of  aqueous  solutions.      (After  Cohen.) 

against  the  pressures  as  abscissas.  It  is  observed  that  the  vis- 
cosity continues  to  decrease  for  all  pressures  up  to  900  atmos- 
pheres, but  the  decrease  becomes  very  slight  at  23°.  In  Fig.  51 
the  ordinates  are  the  same  as  before  but  the  temperatures  are 
plotted  as  abscissas.  It  is  evident  that  the  curves  are  approach- 
ing each  other  and  the  zero  axis,  and  thus  they  indicate  the 
possibility  that  at  some  higher  temperature  the  curves  will  cross 


140  FLUIDITY  AND  PLASTICITY 

the  zero  axis  and  the  viscosity  will  then  increase  with  the  pressure 
as  in  other  liquids. 

In  Fig.  52  the  percentage  change  in  the  time  of  flow  is  plotted 
against  the  pressure  as  in  Fig.  50.  A  saturated  solution  (25.7 
per  cent)  is  seen  to  behave  unlike  pure  water  but  like  other 
liquids  in  that  the  pressure  causes  an  increase  in  the  time  of 
flow.  The  curves  for  other  concentrations  lie  between  those 
for  pure  water  (0  per  cent)  and  those  for  the  saturated  solution. 
The  continuous  curves  represent  measurements  at  14.5°  and  the 
dotted  curves  represent  measurements  at  2°.  From  a  compari- 
son of  these  it  is  evident  that  the  temperature  coefficient  of  the 
percentage  change  in  the  time  of  flow  decreases  rapidly  as  the 
concentration  of  the  solution  is  increased.  This  is  shown  more 
clearly  in  Fig.  53  where  the  temperatures  are  plotted  as  abscissas 
against  the  percentage  change  in  the  time  of  flow  for  a  pressure  of 
600  atmospheres,  i.e.,  the  percentage  change  in  the  time  of  flow 
is  nearly  constant  in  the  most  concentrated  solution.  Further- 
more in  an  8  per  cent  solution  at  a  pressure  of  600  atmospheres, 
the  effect  of  pressure  on  the  time  of  flow  is  zero  at  11°.  Below 
that  temperature,  pressure  decreases  the  time  of  flow ;  above  that 
temperature,  it  increases  it. 

The  relation  of  the  percentage  change  in  the  time  of  flow  at 
600  atmospheres  pressure  as  ordinates  to  the  percentage  con- 
centration as  abscissas  is  indicated  in  Fig.  54.  At  22.5°  the  per- 
centage change  of  the  time  of  flow  is  a  linear  function  of  the 
concentration,  but  at  2°  this  is  no  longer  true,  the  effect  of  the  first 
additions  of  the  salt  to  water  being  much  greater  than  subsequent 
additions.  The  curves  do  not  cross,  hence  the  effect  of  pressure 
in  the  concentrated  salt  solutions  is  greatest  at  the  high  tempera- 
tures even  up  to  the  point  of  saturation.  Cohen  found  the  oppo- 
site to  be  true  of  turpentine,  viz.,  the  effect  of  pressure  is  greatest 
at  low  temperatures.  According  to  Warburg  and  Sachs,  ether 
behaves  like  turpentine  and  benzene  like  sodium  chloride 
solutions. 

Hauser  (1901)  found  that  the  effect  of  pressure  upon  the  vis- 
cosity of  water  continually  decreases  as  the  temperature  is 
raised  until  it  becomes  zero  at  32°  up  to  400  atmospheres.  Above 
this  temperature  the  viscosity  increases  with  the  pressure  as  in 
other  liquids,  and  the  effect  becomes  more  pronounced  as  the 


FLUIDITY  AND  TEMPERATURE  141 

temperature  is  raised,  amounting  to  4  per  cent  for  a  pressure  of 
400  atmospheres  at  100°. 

Faust  (1913),  using  pressures  as  high  as  3,000  kg  per  square 
centimeter,  found  that  the  viscosity  of  ether,  alcohol,  and  carbon 
disulfide  were  each  increased  by  about  fourfold.  This  result 
has  important  bearing  upon  the  theory  and  practice  of  lubrica- 
tion. And  very  recently  J.  H.  Hyde  (1920)  has  reported  to 
the  Lubricants  and  Lubrication  Committee  of  the  Department  of 
Scientific  and  Industrial  Research  the  results  of  an  investigation 
of  the  viscosity  of  a  variety  of  lubricating  oils,  using  pressures 
up  to  7  tons  per  square  inch.  He  made  the  important  deduction 
that  the  mineral  oils  increase  in  viscosity  far  faster  with  the 
pressure  than  do  the  fixed  oils.  Thus  the  viscosity  of  Mobiloil  BB 
increases  over  twenty-six-fold,  whereas  with  the  same  increase 
in  pressure  the  fixed  oils  increase  in  viscosity  about  fourfold. 

Fluidity  and  Volume. — We  have  now  before  us  the  two  follow- 
ing generalizations:  (1)  An  increase  in  pressure  is  usually 
associated  with  a  decrease  in  fluidity,  and  (2)  an  increase  in 
temperature  is  usually  associated  with  an  increase  in  fluidity. 
To  be  sure,  there  are  prominent  exceptions  to  both  generaliza- 
tions, as,  for  example,  water,  in  its  behavior  under  pressure  and 
sulfur,  as  affected  by  temperature.  But  water  and  sulfur  are 
highly  associated  in  the  liquid  state  so  that  an  explanation  of 
these  exceptions  is  possible  on  the  basis  of  changing  molecular 
weights. 

Lowering  of  pressure  or  raising  of  the  temperature  of  a  liquid 
have  one  thing  in  common  in  addition  to  their  similar  effect 
upon  the  fluidity — they  both  produce  an  increase  in  the  volume, 
to  which  there  are  very  few  exceptions.  It  is  worth  while  there- 
fore to  investigate  the  question  of  how  much  of  the  change  in 
fluidity  can  be  attributed  primarily  to  a  change  in  volume.  If 
one  has  in  mind  the  fact  that  in  gases,  where  the  volume  changes 
are  large,  the  fluidity  is  nearly  independent  of  the  volume,  one 
would  naturally  expect  the  changes  in  the  volume  of  liquids  to 
be  responsible  for  only  a  small  part  of  the  fluctuations  in  fluidity 
which  actually  exist.  But  the  viscosities  of  gases  and  liquids 
arise  from  entirely  different  causes,  hence  reasoning  by  analogy 
is  useless. 

The  parallelism  between  fluidity  and  volume  may  be  followed 


142  FLUIDITY  AND  PLASTICITY 

in  another  direction,  for  generally  speaking  whenever  two  liquids 
are  mixed  and  a  contraction  takes  place,  there  seems  to  follow  a 
decrease  in  fluidity.  Alcohol  and  water,  acetic  acid  and  water, 
and  chloroform  and  ether  are  a  few  examples.  On  the  contrary, 
when  liquids  mix  with  an  expansion  in  volume,  the  mixture  has  a 
greater  fluidity  than  we  would  expect  from  the  linear  fluidity-vol- 
ume concentration  curve.  Methyl  iodide  and  carbon  disulphide 
furnish  an  example  of  this  sort  (Bingham  et  al,  1913).  The  above 
facts  have  suggested  to  various  workers  (Brillouin  (1907),  Part 
2,  p.  127;  Dunstan  and  Thole  (1909),  p.  204;  Bingham  (1911), 
p.  270)  that  fluidity  and  volume  are  intimately  related,  more  so 
in  fact  than  fluidity  is  to  either  temperature  or  pressure. 

In  spite  of  this  intimate  relationship,  it  has  been  little  investi- 
gated. Slotte  (1894)  stated  that  the  logarithms  of  the  viscosity 
are  proportional  to  the  logarithms  of  the  specific  volume,  and 
from  this  observation  he  deduced  his  second  Eq.  (43).  But  the 
most  important  discovery  was  made  by  Batschinski  in  1913.  He 
found  that  the  relation  between  the  molecular  volume  V  and  the 
fluidity  may  be  expressed  in  the  following  formula: 

0  =  C(V  -  12)  (56) 

where  12  and  C  are  constants.  The  constant  ft  may  be  defined  as 
the  limiting  value  which  the  molecular  volume  of  any  liquid  can 
have  as  its  fluidity  approaches  zero,  and  it  is  therefore  called 
the  "molecular  limiting  volume."  Consequently  the  difference 
V  —  S2  may  be  called  the  "molecular  free  volume"  and  the  above 
relation  may  be  very  simply  expressed  as  follows:  The  fluidity 
varies  directly  as  the  free  molecular  volume.  Sixty-six  of  the  87 
substances  investigated  by  Thorpe  and  Rodger  exhibit  this  rela- 
tionship and  the  values  of  the  fluidity,  as  calculated  from  the 
volume,  seldom  deviate  from  the  observed  values  by  more  than 
1  per  cent.  The  21  substances  for  which  the.  agreement  is  not 
good  include  the  alcohols,  water,  some  of  the  acids,  the  acid  anhy- 
drides, and  some  of  the  halides.  These  substances  are  generally 
regarded  as  associated  and  it  may  well  be  that  the  molecular 
weight  is  not  constant  as  the  temperature  is  raised.  The  nature 
of  the  very  remarkable  agreement  obtained  is  shown  in  Table 
XXXVII  containing  data  for  benzene  obtained  by  Thorpe  and 
Rodger  between  0°  and  the  boiling-point  and  by  Heydweiller 


FLUIDITY  AND  TEMPERATURE 


143 


from  the  boiling-point  up  to  185.7°.     This  agreement  is  shown 
graphically  for  a  number  of  substances  in  Fig.  55. 

TABLE  XXXVII. — CALCULATION  OP  THE  FLUIDITY  OP  BENZENE  FROM  ITS 
VOLUME    BY    MEANS    OP    THE    FORMULA    <p  =  (V  —  81.76)/0.045,35 


Temperature 

<f>  Observed 

Specific 
volume 

<f  Calculated 

Difference 

0.0 

110.8 

.1124 

111 

0 

10.0 

131.5 

.1242 

132 

0 

20.0 

154.1 

.1377 

155 

1 

30.0 

178.0 

.1514 

179 

1 

40.0 

203.1 

.1661 

204 

1 

50.0 

228.8 

.1812 

230 

1 

60.0 

256.1 

1.1966 

256 

0 

70.0 

284.9 

1.2124 

283 

-2 

80.0 

305.8 

1.2278 

311 

5 

78.4 

314.0 

1.2253 

310 

-4 

100.5 

383.7 

1.2624 

385 

1 

131.8 

504.8 

1  .  3255 

510 

5 

161.4 

646.8 

1.3957 

649 

2 

185.7 

797.4 

1.4661 

794 

-3 

Batschinski  tested  his  formula  with  the  recently  obtained 
data  of  Phillips  (1912)  on  the  viscosity  of  carbon  dioxide  under 
varying  pressures.  He  thus  proved  that  at  least  while  the 
substance  remains  liquid  the  fluidity  varies  directly  as  the  free 
volume. 

TABLE  XXXVIII. — CALCULATION  OP  THE  FLUIDITY  OF  CARBON  DIOXIDE 

Y QQ  2 

FROM  ITS  VOLUME  BY  MEANS  OF  THE  FORMULA  <p  =  ~~~ 


Pressure, 
atmospheres 

Observed 

Specific 
volume 

Calculated 

Per  cent, 
difference 

110.5 

1,299 

1.259 

1.300 

i 

0 

96.0 

1,443 

1.316 

1.450 

1 

82.0 

1,689 

1.397 

1,640 

-3 

76.0 

1,890 

1.471 

1,850 

-2 

72.0 

2,183 

1.575 

2,080 

—  5 

144  FLUIDITY  AND  PLASTICITY 

If  we  could  always  compare  the  fluidities  of  liquids  at  a  definite 
multiple  of  the  molecular  limiting  volume,  it  is  evident  that  the 
effect  of  the  volume,  and  therefore  of  temperature  and  pressure 
in  so  far  as  they  affect  the  volume,  would  be  eliminated.  Such  a 
procedure  would  enormously  simplify  fluidity  relationships. 
As  a  matter  of  fact  Batschinski  has  shown  that  the  molecular 
limiting  volume  possesses  an  additive  character,  the  values  of 


1.18 


50  fluidity  550 

Fio.  55. — Relative  volume-fluidity  curves  after  Batschinski  using  the  data  of 
Thorpe  and  Rodger. 

4,  Bromine;  5,  nitrogen  peroxide;  9,  isohexane;  11,  isoheptane;  13,  trimethyl- 
ethylene;  14,  isoprene;  15,  diallyl;  16,  propylchloride ;  17,  isopropylchloride; 
18,  isobutylchloride ;  19,  allylchloride ;  20,  methylenechloride;  21,  ethylenechlo- 
ride;  22,  ethylidenechloride ;  23,  chloroform;  25,  perchlorethylene ;  68,  ethyl- 
benzene;  69,  orthoxylene;  70,  metaxylene;  71,  paraxylene. 

the  atomic  constants  being  H  =  4.3,  0  =  8.6,  C  =  8.8,  Cl  = 
19.2,  Br  =  24.8,  I  =  32.0,  S  =  19.0,  a  double  bond  =  3.3, 
and  an  iso-grouping  =  0.7.  For  53  of  the  substances  studied  by 
Thorpe  and  Rodger  the  differences  between  the  observed  and 
calculated  values  of  the  molecular  limiting  volume  do  not  exceed 
2  per  cent.  The  limiting  specific  volume  is  approximately  0.307 
of  the  critical  volume  which  is  close  to  the  parameter  b  of  van 
der  Waals'  equation. 

We  are  now  in  a  position  to  get  a  very  clear  understanding 
of  the  law  that  the  fluidity  is  proportional  to  the  free  volume. 


FLUIDITY  AND  TEMPERATURE  145 

When  the  molecules  of  a  liquid  are  closely  packed,  the  volume 
reaches  its  minimum  value  and  the  fluidity  is  zero.  With 
tetrahedral  close-packing  of  the  molecules,  shear  would  require 
rupture  of  the  molecules  themselves.  If  there  are  pore  spaces 
between  or  within  the  molecules,  they  do  not  give  rise  to  fluidity, 
so  that  the  molecules  somewhat  resemble  close-fitting  solid 
figures.  As  the  fluid  expands,  due  to  molecular  agitation,  the 
volume  of  the  molecules  themselves,  i.e.,  the  inner  molecular 
volume  may  remain  the  same,  but  the  ordinary,  i.e.,  the  outer 
molecular  volume  increases.  The  law  states  that  the  fluidity 
originates  solely  in  the  free  space  which  is  the  difference  between 
the  outer  molecular  volume,  or  the  volume  occupied  by  the 
molecules,  and  the  inner  molecular  volume  or  the  space  filled  by 
the  molecules  in  the  sense  indicated  above.  Given  two  sub- 
stances with  the  same  outer  molecular  volume,  it  is  evident  that 
the  one  with  the  larger  molecular  kernel  will  have  the  smaller 
fluidity.  It  is  therefore  natural  to  expect  that  the  limiting 
molecular  volumes  should  be  additive  as  Batschinski  has  found 
to  be  the  case.  This  opens  the  way  to  a  study  of  the  relation 
between  fluidity  and  chemical  composition  and  constitution 
which  is  most  fascinating.  It  is  very  simple  to  measure  the 
outer  molecular  volume,  and  if  this  with  the  fluidity  will  give  a 
certain  and  easy  method  for  determining  the  inner  molecular 
volume,  it  is  a  result  much  to  be  desired.  It  is  apparent  that 
density  and  fluidity  determinations  should  go  hand  in  hand. 
If  the  above  reasoning  held  true  for  gases  as  well  as  liquids, 
the  fluidity  isothermals  of  carbon  dioxide  should  closely  resemble 
the  familiar  volume  isothermals.  By  substituting  for  the  volume 
its  value  in  terms  of  the  fluidity  given  by  Batschinski's  law,  we 
would  obtain  a  modified  van  der  Waals'  equation.  As  a  matter 
of  fact  van  der  Waals'  equation  may  be  written 


V   ___ 
R          R       Rv  ^  Rvz 

but  since  viscosities  are  ordinarily  measured  at  constant,  i.e., 
atmospheric  pressure,  this  may  be  written 


v+D    '    (v+D}* 

where  A,  B,  C,  D  and  #  are  constants.     This  is  identical  with 
Eq.  (54)  which  is  entirely  satisfactory  for  liquids,  except  that  it 


146 


FLUIDITY  AND  PLASTICITY 


coutains  an  additional  term,  which  becomes  negligible  when  the 
fluidity  is  large. 

That  the  fluidity  isothermals  of  carbon  dioxide  do  not  in  any 
way  resemble  the  volume  isothermals  as  we  pass  into  the  gaseous 


<p  x 


10 


FIG.  56. — The  fluidity  isothermals  of  carbon  dioxide  based  on  the  measurements 
of  Phillips. 

condition  is  sufficiently  obvious  from  inspection  of  Fig.  56, 
plotted  from  Phillips'  data.  The  continuous  curves  are  drawn 
between  observed  points.  The  broken  lines  are  added  for 
diagrammatic  purposes.  The  left  half  of  the  figure,  correspond- 
ing to  low  fluidity  and  temperature,  presents  a  strong  similarity 


FLUIDITY  AND  TEMPERATURE  147 

to  the  familiar  pressure-volume  diagram.  At  the  highest 
pressures,  the  fluidity  is  not  greatly  affected  by  a  change  in 
pressure,  e.g.,  at  32°  and  a  pressure  of  120  atmospheres,  a  lowering 
of  the  pressure  by  4  atmospheres  causes  an  increase  in  fluidity 
of  less  than  4  per  cent.  At  a  lower  pressure  the  fluidity  becomes 
extremely  susceptible  to  changes  in  pressure,  a  lowering  of  the 
pressure  by  4  atmospheres  at  76  atmospheres  causing  an  increase 
in  the  fluidity  of  a  full  100  per  cent  at  32°.  The  gaseous  and  liquid 
phases  are  both  present  inside  of  the  curve  kbmcl.  But  the  right  side 
of  the  figure  is  entirely  different  from  the  familiar  pressure- 
volume  diagram.  Instead  of  the  fluidity  being  highly  susceptible 
to  changes  in  pressure,  as  is  the  volume,  it  is  but  slightly  affected, 
e.g.,  at  32°  and  50  atmospheres  pressure,  a  lowering  of  the  pressure 
by  4  atmospheres  causes  only  a  10  per  cent  increase  in  the  fluidity. 

Let  us  follow  in  detail  the  isothermal  of  carbon  dioxide  at  20° 
which  is  well  below  the  critical  temperature.  At  high  pressures, 
the  fluidity  increases  nearly  linearly  from  a  to  6;  there  is  then 
a  sudden  increase  in  the  fluidity  from  1,500  to  5,300  absolute 
units,  as  the  substance  passes  from  the  liquid  to  the  gaseous 
condition.  We  should  expect  the  fluidity  to  continue  to  increase 
as  the  pressure  is  further  lowered,  giving  the  curve  cdf,  but  the 
curve  actually  obtained  is  cd.  We  have  seen  that  the  fluidity  of 
liquids  increases  with  the  temperature,  while,  on  the  other  hand, 
the  fluidity  of  gases  decreases  with  the  temperature,  hence,  the 
pressure-fluidity  curves  for  different  temperatures  must  intersect 
each,  other.  The  figure  proves  that  not  only  is  this  true,  but, 
when  the  temperatures  are  sufficiently  high,  the  curves  all  tend  to 
pass  through  the  particular  point  n,  so  that  at  this  point  the 
fluidity  is  independent  of  the  temperature;  for  the  lower  tempera- 
tures, the  curves  seem  to  intersect  each  other  on  the  curve  ncl. 

Collisional  and  Diffusional  Viscosity.— That  the  pressure- 
fluidity  curves  do  not  follow  an  equation  of  the  van  der  Waals 
type  as  the  fluidity  becomes  large  may  be  due  to  the  appearance 
of  a  new  type  of  viscous  resistance.  We  must  therefore  now 
investigate  more  particularly  into  the  nature  of  viscous  resis- 
tance. One's  first  impulse  in  looking  for  a  cause  of  viscosity  is  to 
assume  a  cohesion  between  the  particles  which  is  exerted  during 
motion  and  acts  in  opposition  to  motion,1  but  with  the  develop- 

1  "Kinetic  Theory  of  Gases,"  MEYER,  p.  171. 


148  FLUIDITY  AND  PLASTICITY 

ment  of  the  kinetic  theory  of  gases,  it  became  evident  that 
viscous  resistance  arises  from  the  diffusion  of  the  particles 
of  high  translational  velocity  into  layers  whose  translational 
velocity  is  lower,  and  vice  versa.  According  to  this  explanation, 
the  loss  of  translational  velocity  must  increase  with  the  tempera- 
ture, which  accords  with  the  fact  that  the  fluidity  of  a  gas 
decreases  as  the  temperature  is  raised. 

But  in  liquids  the  fluidity  increases  with  the  temperature 
and  it  is  generally  agreed  that  there  is  a  second  cause  of  viscous 
resistance,  which,  without  any  very  good  reason  in  its  favor, 
has  been  repeatedly  attributed  to  the  attraction  between  the 
molecules.  According  to  Batschinski l "  If  we  think  of  two  parallel 
layers  of  liquid  as  of  two  rows  of  men,  the  men  moving  in  place 
of  molecules,  we  must  assume  that  these  men  take  hold  of  their 
nearest  neighbors  and  hold  them  for  a  time."  This  explanation  is 
however  inadequate,  for  a  particle  A,  coming  within  the  range 
of  attraction  of  a  particle  B  in  an  adjacent  layer  supposed  to  be 
possessed  of  slightly  less  translational  velocity,  will  be  accel- 
erated and  only  after  passing  B  will  the  retardation  take  place. 
Apparently  the  two  actions  exactly  neutralize  each  other,  or  if 
they  do  not  there  must  result  a  destruction  of  energy  in  violation 
of  the  first  law  of  thermodynamics.  No  reasonable  hypothesis 
has  been  proposed  to  extricate  us  from  this  dilemma,  on  the 
basis  of  cohesion,  hence,  we  are  forced  to  look  for  some  other 
cause.  Whatever  the  explanation,  it  must  show  how  transla- 
tional or  ordered  motion  is  being  continuously  transformed 
into  heat  or  disordered  motion. 

To  get  a  clearer  idea  of  the  nature  of  the  two  causes  of  viscous 
resistance,  we  imagine  two  parallel  planes  A  and  B,  the  former 
moving  to  the  right  parallel  to  itself  in  respect  to  the  second 
plane,  which  for  convenience  only  may  be  assumed  to  be  at  rest. 
We  will  first  assume  that  between'  the  planes  there  is  a  highly 
rarefied  gas.  If  the  walls  are  smooth  and  unyielding  and  the 
particles  of  gas  perfectly  elastic  spheres,  we  will  not  have  a 
model  of  viscous  flow;  for  as  the  particles  collide  with  the  sur- 
faces, the  angle  of  rebound  will  be  equal  to  the  angle  of  incidence, 
there  will  be  no  translational  velocity  transmitted  to  or  from  the 
walls  and  the  so-called  "slipping"  would  be  perfect.  In  order 

1 (1913)  p.  643. 


FLUIDITY  AND  TEMPERATURE 


149 


to  obtain  a  model  of  viscous  flow  it  is  therefore  necessary  to 
assume  that  the  surfaces  are  not  perfectly  smooth.  In  view  of  the 
known  discontinuity  of  matter,  one  could  hardly  assume  a 
smooth  surface,  and  the  least  degree  of  roughness  which  one 
could  well  assume  would  be  one  made  up  of  equal  spheres 
whose  centers  lie  in  the  same  plane  and  as  closely  packed 
together  as  possible.  That  there  is  a  greater  degree  of  roughness 
in  all  ordinary  surfaces  is  probable,  but  it  suffices  for  our  present 
purposes  to  show  in  what  follows  that  this  simple  assumption 
in  regard  to  the  nature  of  the  surfaces  gives  a  workable  model  of 
viscous  flow. 

It  becomes  necessary  to  show  that  momentum  is  being  con- 


M 


N 


PIG.  57. — A  diagram  illustrating  how  traiislational  motion  becomes  changed  into 
vibrational  motion  by  striking  a  rough  surface. 

tinually  taken  from  the  surface  A  and  changed  into  heat.  That 
the  model  meets  the  requirements  depends  upon  the  truth  of  the 
following  theorem:  When  a  series  of  elastic  particles  strike  a 
rough  surface,  the  resultant  component  of  velocity  along  the  surface 
will  be  diminished.  Let  M,  N,  and  P  in  Fig.  57  represent  the 
section  through  the  centers  of  three  of  the  greatly  magnified 
spheres  supposed  to  make  up  the  surface.  It  is  evident  that  if  a 
small  particle  were  to  strike  such  a  surface  at  an  angle  0,  its 
possible  paths  in  striking  the  sphere  N  would  all  lie  between  A  and 
G.  Considering  the  directions  of  the  particle  before  and  after 
collision,  assuming  that  the  angle  of  rebound  at  any  point  of 
the  surface  is  equal  to  the  angle  of  incidence,  we  find  that  for 
possible  paths  between  B  and  D  the  average  resultant  velocity  on 


' 


150  FLUIDITY  AND  PLASTICITY 

rebound  is  exactly  opposite  in  direction,  although  diminished  in 
amount.  For  paths  between  A  and  B  a  particle  would  collide 
with  M  on  rebounding  from  N  but  the  component  of  the  velocity 
in  the  direction  NP  is  diminished.  Also  for  paths  between  D  and 
E,  as  well  as  between  F  and  G,  the  component  of  the  velocity 
in  the  direction  NP  will  be  diminished.  Only  between  E  and  F 
is  the  component  in  the  direction  of  the  flow  greater  after  collision 
than  before.  But  the  distance  EF  becomes  zero  when  6  =  90° 
and  it  has  its  maximum  value  when  6  =  0°,  i.e.,  when  the  trans- 
lational  motion  is  zero.  Since  all  of  the  paths  between  A  and  G 
are  equally  likely,  it  is  clear  that  for  this  section  at  least  the 
average  translational  velocity  is  diminished  by  collision,  irrespec- 
tive of  the  size  of  the  angle  or  of  the  velocity  of  the  particle,  and 
the  same  would  be  true  even  if  the  particle  were  of  considerable 
size.  The  same  must  be  true  a  fortiori  for  sections  other  than  the 
one  passing  through  the  centers  of  the  spheres,  for  then  there 
must,  after  collision,  be  a  component  velocity  at  right  angles 
to  the  plane  of  the  paper  and  therefore  to  the  direction  of  flow. 
The  section  would  be  similar  to  the  one  given  except  that  the 
circles  would  not  touch,  the  spaces  between  them  corresponding 
to  the  pores  of  the  surface  in  which  the  translational  velocity 
would  quite  certainly  be  changed  to  disordered  motion. 

It  follows  from  the  above  that  a  fluid  in  contact  with  a  rough 
surface  tends  to  have  a  translational  velocity  identical  with  that 
of  the  surface1.  Reverting  to  our  model,  the  theorem  explains 
how  molecules  striking  the  surface  A  receive  its  translational 
velocity  and  how  this  translational  velocity  becomes  trans- 
formed into  disordered  motion  at  the  surface  B.  If  the  motion 
of  the  surface  A  were  suddenly  stopped,  all  of  the  flow  would 
cease  in  a  time  which,  for  gases  made  up  of  particles  whose 
velocity  is  expressed  in  kilometers  per  minute,  must  be  quite 
inappreciable.  It  is  to  be  particularly  noted  that  collisions 
between  molecules  of  a  gas  are  unnecessary  for  this  type  of  viscous 
resistance.  This  type  of  resistance  is  caused  solely  by  the  diffu- 
sion of  the  molecules  and  it  therefore  may  be  appropriately 
referred  to  as  diffusional  viscosity. 

For  the  opposite  extreme  we  may  take  for  consideration  a 
very  viscous  liquid.  The  molecular  free  path  is  so  greatly  reduced 

1  Cf.  Jeans  (1904)  and  Dushman  (1921). 


FLUIDITY  AND  TEMPERATURE  151 

that  diffusion  between  adjacent  layers  is  comparatively  slight, 
whereas  the  volume  of  the  molecules  themselves  is  a  consider- 
able portion  of  the  total  volume  of  the  liquid.  Given  a  layer  of 
molecules  C  whose  translational  velocity  is  higher  than  that  of 
another  layer  D,  there  must  of  necessity  occur  collisions  between 
the  two  layers  due  to  the  flow,  and  quite  irrespective  of  any 
diffusion,  provided  only  that  the  diameter  of  the  molecules  is 
greater  than  the  distance  between  the  layers.  On  collision  the 
translational  velocity  is  partly  communicated  to  the  slower 
moving  molecules  of  the  layer  D,  so  that  the  molecules  of  the 
layer  D  have  a  mean  resultant  velocity  in  the  direction  of  the  flow, 
the  remainder  of  the  translational  motion  being  converted  into 
disordered  motion  or  heat.  When  the  system  has  reached  a 
steady  state,  any  layer  D  imparts  to  the  layer  E  below  it  the  same 
amount  of  translational  momentum  that  it  has  received  from  the 
layer  C  above  it,  except  for  the  amount  of  energy  which  is  being 
continuously  changed  into  heat,  and  it  is  this  disappearance  of 
translational  momentum  which  gives  rise  to  the  new  type  of 
viscous  resistance  known  as  collisional  viscosity.  Since  each 
layer  is  able  to  transmit  but  a  portion  of  the  translational  momen- 
tum which  it  receives  to  the  adjacent  more  slowly  moving  layers, 
there  results  a  steady  diminution  in  the  velocity  of  flow  from  the 
most  rapidly  moving  layer  A  to  the  layer  which  is  at  rest  B. 

From  this  model  of  viscous  flow  in  liquids  it  is  possible  to  deduce 
the  effects  of  changes  in  concentration,  pressure,  temperature,  and 
size  of  the  molecules.  The  number  of  collisions  of  the  particles 
of  one  layer  with  those  of  another  layer,  due  to  translational 
velocity,  will  be  directly  proportional  to  the  rate  of  shear  between 
the  layers.  This  is  the  fundamental  law  of  viscous  flow.  It  will 
also  be  directly  proportional  to  the  number  of  molecules  in  each 
layer,  i.e.  to  the  concentration.  It  is  a  confirmation  of  this 
prediction,  that  we  find  that  the  fluidity  is  decreased  in  just  the 
proportion  that  the  concentration  is  increased  either  by  lowering 
the  temperature  or  by  raising  the  pressure.  It  is  significant 
that  the  temperature  by  itself  is  without  effect  on  collisional  vis- 
cosity. The  reason  for  this  is  evidently  that  the  mere  vibration 
of  the  molecules  without  diffusion  through  successive  layers  does 
not  affect  the  rate  at  which  the  molecules  of  one  layer  overtake 
the  molecules  of  an  adjacent  layer  moving  more  slowly.  It  is 


152  FLUIDITY  AND  PLASTICITY 

clear  that  collisional  viscosity  will  increase  not  only  with  the 
concentration  but  also  with  the  size  of  the  molecules.  If  the 
particles  were  mere  points,  there  would  be  no  collisions  and 
therefore  no  collisional  resistance  to  flow.  On  the  other  hand,  if 
the  molecules  completely  filled  the  space  they  occupy,  collisions 
would  be  most  rapid  and  the  collisional  resistance  a  maximum. 

The  discovery  of  Batschinski,  that  in  unassociated  liquids 
the  fluidity  is  directly  proportional  to  the  free  volume,  seems  to 
indicate  that  collisional  viscosity  is  almost  entirely  responsible 
for  the  viscosity  of  ordinary  liquids  and  it  must  be  highly  impor- 
tant in  compressed  gases.  It  is  also  clear  why  associated  liquids 
are  exceptional.  For  the  breaking  down  of  association,  as  by 
heating,  would  doubtless  decrease  the  size  of  the  molecules  without 
a  corresponding  decrease  in  the  space  which  they  occupy. 

The  Mixed  Regime.  —  It  has  been  indicated  that  in  rarefied 
gases  viscous  resistance  is  certainly  diffusional  and  in  very  viscous 
liquids  it  is  collisional.  In  fluids  at  ordinary  temperatures  and 
pressures  the  viscous  resistance  is  evidently  the  sum  of  the 
diffusional  and  the  collisional  resistances.  The  total  viscous 
resistance  is  in  every  case  given  by  the  equation 

fl  =  rid  +  r)c  (57) 

where  rjd  is  the  diffusional  viscosity  and  ijc  is  the  collisional 
viscosity. 

According  to  Maxwell,  as  discussed  in  Chapter  XIV,  the 
viscosity  of  a  gas  varies  as  the  absolute  temperature,  so 


where  B  is  a  constant.  Later  experimenters  have  found  that  this 
formula  does  not  accord  with  the  experimental  facts,  and  they 
have  therefore  given  to  the  temperature  T  an  exponent  n  with 
values  varying  from  the  theoretically  deduced  0.5  to  1.0.  The 
discrepancy,  however,  may  be  due  to  the  fact  that  collisional  vis- 
cosity has  been  overlooked.  For  diffusional  viscosity  we  here 
assume  as  a  first  approximation  that  n  =  I. 

We  have  seen  that  Batschinski's  formula  represents  collisional 
viscosity  only,  which  we  may  now  write  in  the  form 
A 


FLUIDITY  AND  TEMPERATURE 


153 


TABLE  XXXIX. — THE  FLUIDITY  OF  CARBON  DIOXIDE  AS  CALCULATED  BY 

MEANS  OF  THE  FORMULA  <p  =  (v  —  u)[A  +  BT  (v  —  «)]  WHERE  w  = 

0.841,  A  =  0.000,257,8,  AND  B  =  4,998  COMPARED  WITH  THE 

VALUES  OBSERVED  BY  PHILLIPS  AT  VARIOUS  TEMPERATURES 

AND  PRESSURES 


Temperature, 
absolute 

Pressure  in 
atmospheres 

r 

<f>  observed 

<p  calculated 

Per  cent, 
difference 

293 

83.0 

.198 

1,215 

1,152 

_ 

72.0 

.232 

1,297 

1,241 

-    4 

59.0 

.302 

1,435 

1,418 

-    1 

56.0 

.263 

5,376 

4,890 

-   9 

50.0 

.897 

5,650 

5,299 

-   6 

40.0 

1   .000 

6,024 

5,734 

-    5 

20.0 

2  .780 

6,410 

6,421 

0 

1.0 

54   .400 

6,757 

6,907 

+   2 

303 

110.5 

.258 

1,299 

1,300 

0 

104.0 

.280 

1,364 

1,355 

-    1 

96.0 

.316 

1,443 

1,441 

0 

90.0 

.346 

1  ,  555 

1,512 

-   3 

82.0 

.397 

1,689 

1,519 

-11 

80.0 

.416 

1,770 

1,668 

-    6 

76.0 

.471 

1,890 

1,784 

-    6 

74.0 

.506 

2,020 

1,855 

-    8 

73.0 

.531 

2,092 

1,906 

-    9 

72.0 

.575 

2,183 

1,992 

-   9 

70.0 

3.484 

4,367 

4,021 

-   8 

60.0 

5.650 

5,348 

4,887 

-    9 

40.0 

10.  870 

5,952 

5,654 

-    5 

20.0 

28.250 

6,289 

6,086 

-    3 

1.0 

565.000 

6,536 

6,594 

+    1 

305 

120.0 

.266 

1,269 

1,318 

+   4 

112.0 

.287 

1,350 

1,369 

+    1 

104.0 

.316 

1,439 

1,439 

0 

93.0 

.372 

1,595 

1,568 

-    2 

87.0 

.429 

1,706 

1,693 

-    1 

84.0 

.466 

1,786 

1,771 

-    1 

80.0 

.527 

1,894 

1,894 

0 

76.0 

.675 

2,232 

2,168 

-   3 

75.0 

.802 

2,463 

2,379 

-    3 

74.0 

2.778 

3,937 

3,506 

-    8 

70.0 

3.922 

4,673 

4,240 

-    9 

60.0 

5.882 

5,348 

4,918 

-    8 

40.0 

11.111 

5,714 

5,641 

-    1 

20.0 

28.410 

6,173 

6,201 

0 

1.0 

568.  100 

6,452 

6,553 

+   2 

308 

114.5 

1.324 

1,443 

1,455 

+    1 

109.0 

1.349 

1,515 

1,512 

0 

96.0 

1.437 

1,706 

1,706 

0 

88.0 

1.531 

1,957 

1,896 

-    3 

85.0 

1.597 

2,193 

2,021 

-    8 

80.0 

2.024 

2,770 

2,690 

-    3 

75.0 

3.460 

4,219 

3,966 

-    6 

70.0 

4.405 

4,673 

4,425 

-    5 

60.0 

6.135 

5,618 

4,943 

-12 

40.0 

11.765 

5,747 

5,642 

-    2 

20.0 

28.  740 

6,135 

6,135 

0 

1.0 

574.  700 

6,410 

6,487 

+    1 

313 

112.0 

1.431 

1,751 

1,686 

-    4 

108.0 

1.466 

1,852 

1,759 

-    5 

100.0 

1.572 

2,070 

1,965 

- 

94.0 

1.718 

2,415 

2,221 

-    8 

85.0 

2.597 

3,717 

3,302 

-13 

80.0 

3.436 

4,587 

3,920 

-14 

70.0 

4.902 

5,000 

4,553 

-    9 

60.0 

6.536 

5,348 

4,964 

-    7 

40.0 

12.050 

5,682 

5,585 

—    2 

23.8 

24.510 

5,917 

5,987 

+    1 

1.0 

578.000 

6,369 

6,385 

0 

154  FLUIDITY  AND  PLASTICITY 

where  A  and  w  are  constants  and  v  is  the  specific  volume  in  ml 
per  gram.     We  have  then 

rj  =  RT  +—  - 


y  (58) 

It  is  truly  remarkable  that  so  simple  an  equation  as  this 
can  be  employed  with  success  to  reproduce  so  complex  data  as 
that  for  the  fluidity  isothermals  of  carbon  dioxide  passing  through 
the  critical  state.  To  what  extent  it  does  do  this  is  shown  in 
Table  XXXIX.  Since  the  calculated  values  are  nearly  all  too 
small,  it  is  evident  that  a  better  concordance  could  have  been 
secured  by  a  happier  choice  of  constants,  but  considering  the 
difficulties  in  these  measurements,  the  percentage  of  deviation 
between  the  calculated  and  the  observed  values  is  not  large. 

Having  established  a  fairly  exact  relationship  between  fluidity 
and  volume,  and  indirectly  with  temperature  and  pressure, 
the  problem  of  associated  substances  again  presses  itself  into 
the  foreground  as  it  tends  to  do  so  often.  A  means  must  be 
found  for  bringing  these  substances  into  conformity  with  the 
others,  but  the  solution  is  not  yet  forthcoming. 

Dr.  Kendall  inquires  in  regard  to  the  foregoing:  —  "Is  the 
formula  of  Batschinski  of  such  great  importance  as  your  extended 
treatment  of  it  would  lead  the  readers  to  believe?  Is  it  not 
merely  an  interpolation  formula?  Would  it  not  be  well  to 
mention  something  about  the  alternative  formula  of  Arrhenius?" 
(1918).  The  expotential  formula  of  Arrhenius  (1918)  does  not 
lead  us  to  a  definite  mental  picture,  and  like  many  another 
frankly  empirical  formula  was  omitted  in  this  brief  treatment 
of  the  subject.  The  relation  of  Batschinski  fills  a  need  which 
was  felt  in  many  minds,  cf.  p.  142.  It  leads  us  at  once  to  a 
definite  mental  picture  which  is  neccessary  in  building  up  a 
consistent  theory,  so  that  we  are  now  able  to  explain  the  relation 
of  fluidity  to  volume,  temperature  and  pressure  et  cet.  in  a 
manner  which  is  so  natural,  so  unexpectedly  simple  and  so  beau- 
tifully in  accord  with  observed  facts  that  it  is  hard  to  see  what 
more  evidence  is  needed  to  carry  conviction. 


CHAPTER  IV 
FLUIDITY  AND  VAPOR  PRESSURE 

All  physical  and  chemical  properties  will  perhaps  in  time 
be  shown  to  be  related  so  that  the  knowledge  of  a  certain  set  of 
facts  in  regard  to  a  substance,  such  for  example  as  its  chemical 
structure,  will  enable  one  to  deduce  it  multitudinous  properties. 
Thus  having  established  a  direct  causal  dependence  of  the  fluidity 
upon  the  volume,  it  is  also  important  to  study  other  properties 
which  depend  upon  the  fluidity,  or  which  together  with  the 
fluidity  depend  upon  a  common  cause.  Migration  velocity  and 
electrical  conductivity  of  solutions  are  examples  of  properties 
which  are  directly  dependent  upon  the  fluidity.  There  are 
properties  which  are  not  dependent  upon  the  fluidity  directly  but 
which  with  the  fluidity  are  dependent  upon  the  same  property 
and  therefore  are  indirectly  related.  The  boiling-point,  the 
critical  temperature  and  the  vapor-pressure  are  properties  of  this 
latter  type,  which  we  will  now  consider. 

Fluidity  and  Boiling-point. — On  examination  of  the  fluidity- 
temperature  curves  of  the  aliphatic  hydrocarbons,  Fig.  41,  and 
ethers,  Fig.  42,  we  note  that  the  fluidities  of  these  substances  at 
their  boiling-temperatures — shown  by  small  circles — are  nearly 
identical.  It  is  perhaps  of  no  special  significance  that  the  flu- 
idities are  identical,  but  it  is  important  that  the  line  connecting 
the  fluidities  at  the  boiling-points  is  linear.  This  linear  character 
of  the  fluidity-boiling-point  curve  is  exemplified  by  the  aliphatic 
chlorides,  bromides  and  iodides  as  well  as  by  the  ethers  and 
hydrocarbons.  The  acids  and  alcohols  are  again  exceptional. 

The  meaning  of  this  relation  may  be  most  easily  grasped 
by  reference  to  Fig.  42.  If  we  assume  that  the  curves  of  the 
members  of  a  given  class  have  the  same  slope  and  the  sajne 
degree  of  curvature  at  the  boiling-point,  it  is  evident  that  the 
addition  of  a  methylene  group  to  a  molecule  causes  a  rise  in  the 
boiling-point  F  measured  by  AC1  or  CE,  but  at  the  same  time 

1  Cf.  Fig.  42. 

155 


156 


FLUIDITY  AND  PLASTICITY 


the  addition  of  a  methylene  group  causes  a  decrease  in  the  fluidity 
A  which  is  AB  or  CD.  It  appears  that  AC  JAB  =  CE/CD,  so 
the  ratio  between  the  effect  produced  on  the  boiling-point  to  the 
effect  produced  on  the  fluidity  T/A  is  constant,  for  this  particular 
homologous  series.  This  relation  does  not  apply  to  the  alcohols 
and  acids.  A  reason  is  that  as  the  temperature  is  raised,  the 
association  is  lowered,  F  becomes  smaller  and  at  the  same  time  A 
becomes  greater,  so  that  their  ratio  may  vary. 

Fluidity  and  Vapor  Pressure. — If  there  is  a  relation  between 
the  fluidity  and  the  boiling-point,  it  is  evident  that  a  more 


400 


300 
200 
100 


Methyl  Propul 
Ethyl  Propyl  E 


Ether 
fj/ Ether 
Dipropul  Ether 
Diethyl  Ether- 


0        100      200      300      400       500      600      TOO      800      900      1000 

P 
FIG.  58. — Fluidity-vapor  pressure  curves  of  a  series  of  ethers. 


general  relation  than  the  above  can  be  obtained  by  comparison 
at  other  vapor  pressures  than  at  the  ordinary  boiling  temperature. 
Thus  if  all  of  the  aliphatic  ethers  have  the  same  fluidity  at  the 
ordinary  boiling-point,  and  the  same  were  true  at  other  vapor 
pressures,  it  follows  that,  when  the  vapor  pressures  corresponding 
to  a  given  temperature  are  plotted  against  the  fluidities  corre- 
sponding to  that  temperature,  and  this  process  is  repeated  for  a 
series  of  temperatures,  the  curves  of  all  of  the  substances  of  the 
class  should  fall  together;  in  other  words,  the  fluidity  vapor-pressure 
curve  of  one  ether  ought  to  be  the  curve  of  all  the  other  members 
of  the  class.  Conversely,  if  either  the  vapor  pressure  or  the 
fluidity  of  an  ether  is  known  for  a  given  temperature,  the  other 
quantity,  supposedly  unknown,  can  be  determined  by  means  of 


FLUIDITY  AND  VAPOR  PRESSURE 


157 


the  fluidity  vapor-pressure  curve  of  the  class.  Not  only  do  all 
the  members  of  this  class  fall  together  in  a  single  parabolic  curve, 
shown  in  Fig.  58,  but  substances  of  other  classes  give  a  curve  of 
similar  form,  as  will  now  be  demonstrated. 

If  we  take  a  single  substance,  such  as  heptane,  as  our  standard 

TABLE  XL. — FLUIDITIES  AND  VAPOK  PRESSURES  CORRESPONDING  TO   THE 
STANDARD  FLUIDITY  VAPOR-PRESSURE  CURVE 


Fluidity 
reduced 

Vapor 
pressure 
in  mm 

Fluidity 
reduced 

Vapor 
pressure 
in  mm 

Fluidity 
reduced 

Vapor 
pressure 
in  mm 

100 

0.5 

310 

111.1 

420 

396.7 

110 

0.7 

315 

119.0 

425 

415.7 

120 

1.0 

320 

127.8 

430 

434.9 

130 

1.7 

325 

136.5 

435 

455.5 

140 

2.2 

330 

146.1 

440 

476.1 

150 

3.0 

335 

155.4 

445 

497.3 

160 

4.3 

340 

165.2 

450 

520.0 

170 

6.1 

345 

175.5 

455 

542.2 

180 

8.2 

350 

186.2 

460 

566.8 

190 

11.0 

355 

197.6 

465 

589.5 

200 

14.2 

360 

210.4 

470 

614.7 

210 

17.7 

365 

223.2 

475 

635.9 

220 

22.3 

370 

236.6 

480 

660.1 

230 

27.7 

375 

251.0 

485 

683.7 

240 

34.1 

380 

266.0 

490 

711.7 

250 

41.2 

385 

281.1 

495 

736.3 

•  260 

49.3 

390 

296.2 

500 

760.0 

270 

58.8 

395 

311.4 

505 

783.8 

280 

69.1 

400 

328.3 

510 

810.0 

290 

81.8 

405 

344.7 

515 

838.8 

300 

95.9 

410 

360.7 

520 

869.5 

305 

103.2 

415 

378.4 

525 

902.0 

substance  with  a  fluidity  of  approximately  500  at  the  boiling- 
point,  we  can  compare  other  substances  with  this  one.  Since 
other  substances  do  not  have  the  same  fluidity  at  the  boiling- 
point,  we  multiply  the  fluidity  at  the  boiling-point  by  a  factor 
so  that  the  product  or  the  reduced  fluidity  will  be  500.  The 
fluidities  for  other  temperatures  are  also  reduced  by  similarly 


158 


FLUIDITY  AND  PLASTICITY 


multiplying  by  the  same  factor.     Since  the  fluidity  vapor-pressure 
curves  pass  through  the  origin  and  we  by  this  process  bring  them 


O  ZOO  200  300  400  500 

Fluidity 

•piQ.  59. — The   vapor-pressure  curves  of  some  associated  substances  compared 
with  heptane. 

together  at  the  boiling-point,  a  comparison  of  the  reduced  values 
for  other  temperatures  with  the  values  for  heptane  will  show 
how  nearly  similar  are  the  curves  for  different  substances,  even 


FLUIDITY  AND  VAPOR  PRESSURE  159 

when  they  belong  to  very  different  classes.  For  17  substances 
compared  by  this  method,  the  average  deviation  between  the 
observed  and  calculated  values  is  approximately  3  per  cent, 
so  long  as  the  vapor  pressure  is  above  10  cm. 

A  few  examples  will  serve  to  make  the  use  of  the  method  clear. 
Thus  the  fluidity  of  ethyl  acetate  at  its  boiling-point  (77.2°C) 
is  395.6  and  the  reduction  factor  is  therefore  500/395.6  =  1.264. 
The  fluidity  of  the  substance  as  determined  by  Thorpe  and 
Rodger  for  30°  is  249.9  and  the  reduced  fluidity  is  therefore 
249.9  X  1.264  =  315.8,  and  the  vapor  pressure  read  from  the 
Table  XL  for  the  standard  curve  corresponding  to  this 
fluidity  is  120.4  mm,  while  the  vapor  pressure  observed  by  Young 
is  118.7  mm.  As  a  further  example,  ethyl  propyl  ether  has  a  flu- 
idity of  479.9  at  its  boiling-point  (63.4°).  The  factor  is  500/479.9 
=  1.042.  The  fluidity  at  20°  as  determined  by  Thorpe  and 
Rodger  is  314.9  and  this  reduced  is  314.9  X  1.042  =  328.1,  and 
according  to  the  Table  XL  this  corresponds  to  142.5  mm  which 
is  practically  identical  with  the  experimental  value  of  142.6. 

Associated  substances  do  not  show  the  relation  between 
fluidity  and  vapor  pressure  shown  elsewhere.  The  greatest 
deviation  is  shown  by  isobutyl  alcohol  and  formic  acid  and  ethyl 
alcohol.  In  Fig.  59,  there  is  plotted  the  vapor-pressure-tempera- 
ture curve  of  heptane  (not  reduced)  with  several  "associated" 
substances  which  show  large  deviation  from  the  standard.  We 
note  that  isobutyl  alcohol,  formic  acid,  and  ethyl  alcohol  show 
the  most  rapid  increase  in  the  vapor  pressure.  This  is  added 
evidence  of  the  breaking  down  of  association.  (C/.  pp.  276 
and  277.) 

Before  concluding  our  consideration  of  vapor  pressure  it  may 
be  remarked  that  since  fluidity  is  related  to  volume  and  at  the 
same  time  to  vapor  pressure,  there  is  necessarily  a  relation  be- 
tween volume  and  vapor  pressure.  The  volume  is  doubtless 
affected  by  the  internal  forces  between  the  molecules  which  we 
ordinarily  call  cohesion  and  vapor  pressure  naturally  depends 
upon  the  same.  So  there  may  quite  possibly  be  a  connection 
between  fluidity  and  cohesion,  even  though  it  is  not  the  connec- 
tion which  is  often  supposed.1 

Cf.  p.  147  et  seq. 


CHAPTER  V 
THE  FLUIDITY  OF  SOLUTIONS 

The  fluidity  curves  of  solutions  are  most  logically  considered 
under  four  types:  I.  In  the  simplest  case  the  fluidity  of  the 
mixture  can  be  calculated  from  the  fluidities  of  the  components. 
There  is  no  volume  change  on  mixing,  and  it  is  assumed  that  the 
components  neither  dissociate  nor  interact  with  each  other  on 
mixing.  The  method  of  calculation  of  the  fluidity  of  the  ideal 
mixture  has  been  the  subject  of  much  discussion  and  it  will  be 
discussed  presently.  Examples  of  this  simplest  type  are  carbon 
tetrachloride  and  benzene,  and  diethyl  ether  and  benzene. 

Thorpe  and  Rodger  (1897)  found  that  there  was  a  very  slight 
contraction  on  mixing  carbon  tetrachloride  and  benzene,  thus 
confirming  the  earlier  observation  of  F.  D.  Brown.  In  the  case 
of  methyl  iodide  and  carbon  disulfide  there  was  a  very  slight 
expansion  which  decreased  as  the  temperature  was  raised. 
Ramsay  and  Aston  found  that  the  surface  tension  of  mixtures  of 
carbon  tetrachloride  and  benzene  followed  the  mixture  rule. 
Zawidski  furthermore  observed  that  the  vapor  pressures  of  these 
same  mixtures  showed  but  a  slight  deviation  from  the  mixture 
rule,  due,  according  to  Dolazalek,  to  association  of  the  carbon 
tetrachloride.  This  is  the  sort  of  parallelism  which  needs  much 
further  investigation  because  it  affords  the  most  nearly  indis- 
putable evidence  to  aid  the  investigator  in  the  selection  of  ideal 
mixtures.  In  much  of  our  physico-chemical  reasoning,  it  would 
beyond  any  question  be  a  great  advantage  if  we  could  assume 
certain  mixtures  as  ideal  in  the  sense  defined  above. 

The  fluidity-volume  concentration  curves  of  this  class  are 
nearly  but  not  quite  linear,  as  will  be  explained. 

II.  There  are  instances  where  there  is  a  well-defined  expansion 
on  mixing,  accompanied  with  heat  absorption,  and  in  such  mix- 
tures we  generally  find  the  fluidity  greater  than  calculated.  The 
fluidity-volume  concentration  is  convex  upward,  i.e.,  the  curva- 
ture d*<f>/dc2  is  negative.  The  increase  in  fluidity  may  be 

160 


THE  FLUIDITY  OF  SOLUTIONS  161 

attributed  to  breaking  down  of  association  or  to  dissociation 
which  also  give  rise  to  the  increase  in  volume.  Benzene  and 
ethyl  acetate  may  be  cited  as  an  example  of  this  type. 

III.  When  two  liquids  are  mixed,  perhaps  more  often  than 
not,  there  is  a  decrease  in  the  volume,  particularly  in  aqueous 
solutions.     With  this  decrease  in  volume  there  goes  a  positive 
heat  effect  and  a  decrease  in  the  fluidity,  so  the  fluidity-volume 
concentration    curves    are    convex    downward,    i.e.,    d2<p/dc2   is 
positive.     Since  the  fluidity  changes  some  2,000  times  as  rapidly 
as  the  free  volume,  as  shown  by  Batschinski's  law,  the  effects 
of  the  solvation  which  is  presumed  to  be  present  in  this  case 
manifest  themselves  far  more  prominently  in  the  fluidity  data 
than  in  the  data  on  volume.     Examples  of  this  type  are  very 
common,  making  up  the  greater  portion  of  aqueous  mixtures, 
such  as  ethyl  alcohol  and  water,  acetic  acid  and  water  and  many 
mixtures  of  organic  liquids  such  as  chloroform  and  ether.     In 
some  cases  there  is  incontestible  proof  that  a  chemical  compound 
is  formed  on  mixing,  as  when  aniline  is  mixed  with  acetic  acid. 
In  other  cases,  the  formation  of  hydrates  or  solvates  is  very 
probable.     Whether  there  is  a  sharp  line  to  be  drawn  between  the 
forces  of  cohesion  of  a  purely  physical  nature  and  the  forces 
which  bring  about  actual  chemical  combination  must  be  decided 
by  future  experiment. 

IV.  When   associated   solvents   break   down,   to   later  unite 
with  each  other,  we  have  a  combination  of  the  second  and  third 
types.     The  resulting  curve  may  then  show  positive  curvature 
over  part  of  its  course  and  negative  curvature  over  a  part, 
there  being  a  point  of  inflection.     Examples  of  this  type  are 
found  in  both  aqueous  and  non-aqueous  solutions,  ammonium 
nitrate  solutions  being  a  good  example  of  the  former. 

Instances  of  this  type  are  instructive,  since  they  put  us  on  our 
guard  against  assuming  that  because  a  given  mixture  displays 
strong  positive  curvature,  dissociation  is  not  a  factor.  A  pair 
of  liquids  may  fall  into  type  II  and  yet  have  a  tendency  to  unit.e 
together  chemically,  provided  merely  that  the  effect  of  dissocia- 
tion predominates  in  all  mixtures. 

Again  the  opposing  effects  may  be  nearly  equal  at  all  concentra- 
tions, as  is  true  of  ethyl  alcohol  and  acetone,  and  formic  acid  and 
water.  These  mixtures  evidently  fall  under  type  IV  and  not 


162 


FLUIDITY  AND  PLASTICITY 


type  I,  but  in  cases  of  doubt  resort  may  be  had  to  a  study  of 
the  other  physical  properties. 

I.  THE  IDEAL  MIXTURE 
If  there  is  no  contraction  on  mixing  two  liquids 


| 

/ 

/ 

7 

Spec  if  ic  Vo 

y 

: 

A 

//   u~ 

ddttive  Sp.Vol 
Dbsewed 

Calc.byWt.° 

0 

25                    50                    75                    IOC 
enzene  WeigW  Concentration  of  Ettier                  Efher 

FIG.  60. — Specific  volume-weight  concentration  curve  of  mixtures  of  benzene 
and  ether.     (After  D.  F.  Brown.) 


v  =  mvi  -f- 


(59) 


p  dpi   -|-  Opa 

where  v  is  the  specific  volume  and  p  the  density  of  the  mixture, 
containing  ra  weight  fraction  or  a  volume  fraction  of  the  com- 
ponent A  whose  specific  volume  is  vi  =  —  and  n  weight  frac- 
tion or  b  volume  fraction  of  the  component  B  whose  specific 

From  the  above  equa- 


„. 

volume  is  t>2  =  —      So  a 
p2 


mvi  + 


THE  FLUIDITY  OF  SOLUTIONS 


163 


tion  it  follows  that  if  we  plot  volumes  against  weight  concentra- 
tion, we  will  obtain  a  linear  curve  such  as  curve  I  in  Fig.  60; 
but  if  we  plot  specific  volumes  against  volume  concentrations, 
we  will  obtain  not  the  linear  curve  III,  Fig.  61,  but  curve  IV. 

We  have  seen  that  the  fluidity  of  a  liquid  is  directly  propor- 
tional to  its  free  volume,  but  the  fluidities  are  additive  (Eq.  (25)) 


1.380 


1.200 


-  Add  itive  Sp.Vol .  Ca  I  c .  by  Vol  .°/o 
BT-Additive  Sp.Vol.Calc.  by  Wt.%> 
V- Observed 


Benzene 


25  50  15  100 

Volume  Concentration  Ether  Ether 


FIG.  61.  —  Specific  volume-volume  concentration  curve  of  mixtures  of  benzene 
and  ether.     (After  D.  F.  Brown.) 


only  when  we  use  volume  percentages;  hence  it  follows  that  if  a 
pair  of  liquids  on  mixing  gave  a  linear  specific  volume-volume 
concentration  curve  (curve  III)  they  would  also  give  a  linear 
fluidity-volume  concentration  curve,  curve  VI,  Fig.  62.  Since, 
however,  the  ideal  mixture  gives  a  volume-volume  concentration 
curve  which  shows  positive  curvature,  the  fluidity-volume  con- 
centration curve  of  the  ideal  mixture  will  also  show  positive 
curvature,  curve  VII,  Fig.  62. 


164 


FLUIDITY  AND  PLASTICITY 


Since  this  sag  in  the  fluidity  curve  is  due  to  the  mathematical 
necessities  of  the  case  and  not  to  chemical  combination  or 
dissociation,  it  is  evidently  possible  to  calculate  the  fluidity  of 
the  mixture  from  the  fluidities  and  volumes  of  the  components. 


400 

// 

350 

/ 

// 

-'/ 

vin 

/ 

lffvx   > 

// 

/   A 

/ 

$ 

'-5 

3    250 

/ 
/ 

/A 

/  ft* 
/ 

ix.     d 

/ 
/ 
/  / 

/  M"c 
vn- 

Additive  Fluic 
alculatedby\ 

Huidity  Gala 
?y  Formula 

)bserved  Poin- 

>ty 
roi°/0 

jla+pd 

150' 

// 

//' 
// 
y 

0     1 

^& 

0                   25                    50                   75                  IOC 
Benzene      v0|limpfnnr-Mfr-+:rt^  mrpu,^               E+he 

FIG.  62.  —  Fluidity-volume  concentration  curve  of  mixtures  of  benzene  and  ether. 
(After  D.  F.  Brown.) 

We  have  seen  that  the  observed  specific  volume  of  the  mixture 


whereas  the  specific  volume  should  be 


in  order  to  give  a  linear  fluidity-volume  concentration  curve 
(Eq.  (25)),  so  the  specific  volume  is  too  small  by  an  amount 
represented  by  the  specific  volume  difference,  At>. 


THE  FLUIDITY  OF  SOLUTIONS  165 

Ay  =  avi  +  bv%  —  mv\  —  nvz 
=  (a  —  m)v\  —  (n  —  6)y2 
=  (a  -  m)  (»!  -  »,)  (60) 

since  a  —  w  =  n  —  b. 

If  the  fluidity  is  directly  proportional  to  the  free  volume  (Eq.  56), 
it  seems  reasonable  to  assume  that  if  the  volume  is  decreased 
for  any  reason  by  an  amount  A»,  the  fluidity  will  be  decreased 
by  an  amount  which  is  some  function  of  this  fAv.  Since  in  the 
ideal  mixture  the  fluidity  is  only  slightly  less  than  that  given 
by  the  linear  formula  (Eq.  25),  we  may  assume  as  a  first  approxi- 
mation that  the  decrease  in  fluidity  is  directly  proportional  to  the 
specific  volume  difference.  We  then  obtain  as  our  formula  for 
the  true  fluidity  $ 

3>  =  k(v  —  w)  —  K&v 
=  s?  -  KAv 
=  dpi  +  b(f>2  —  K(a  —  m)  (v\  —  v%),  (61) 

It  may  be  possible  later  to  evaluate  the  above  function,  but 
it  is  only  necessary  to  know  the  fluidity  of  one  or  more  mixtures 
in  order  to  determine  a  value  for  the  constant  K,  from  which  the 
fluidities  of  all  other  mixtures  of  the  two  components  may  be 
calculated.  The  physical  significance  of  K  will  be  explained  later. 

We  may  take  for  an  example  carbon  tetrachloride  and  benzene, 
the  mixtures  of  which  were  studied  carefully  by  Thorpe  and  Rodger, 
and  at  a  single  temperature  by  Linebarger.  (C/.  Table  XLII.) 
In  the  first  line  at  each  temperature  are  given  the  fluidities 
observed.  In  the  second  line  are  given  the  fluidities  (*)  as 
calculated  with  the  use  of  Eq.  (61),  using  40  as  the  value  for  K. 
The  fluidities  (<p)  as  given  by  the  simple  additive  formula, 
Eq.  (25),  are  given  in  the  third  line.  These  last  are  invariably 
higher  than  the  observed  values,  but  when  corrected  for  the 
volume  the  agreement  is  very  close,  the  average  deviation 
being  less  than  half  of  1  per  cent. 

It  is  worth  noting  in  passing  to  what  extent  Batschinski's  Law 
applies  to  the  ideal  mixture,  with  which  we  assume  to  be  dealing 
in  this  case.  Using  the  limiting  specific  volumes  of  carbon 
tetrachloride  and  benzene  as  0.5782  and  1.0476  respectively, 
we  have  calculated  the  values  of  o>  by  the  admixture  rule  to  be 
0.683,  0.784,  and  0.896,  using  weight  percentages.  This  accords 


166 


FLUIDITY  AND  PLASTICITY 


perfectly  with  the  values  0.683,  0.785,  and  0.897  obtained  from 
the  observed  fluidity  data  for  each  mixture.  The  values  of  k 
in  the  Batschinski  formula  are  2,019,  1,937,  1,845  as  calculated 
from  the  pure  solvents,  as  compared  with  2,034,  1,993,  and  1,876 
as  obtained  from  the  data  for  the  mixtures.  The  values  of 
the  fluidities  calculated  by  means  of  the  former  set  of 
constants  are  not  so  close  to  the  observed  values  as  are  the  values 
calculated  by  the  corrected  fluidity  formula.  But  they  are  at 
least  as  close  to  the  observed  values  of  the  mixtures  as  the  cal- 
culated fluidities  of  pure  carbon  tetrachloride  are  to  the  observed. 
It  is  impracticable  here  to  consider  in  detail  all  of  the  examples 


TABLE  XLI. — SPECIFIC  VOLUMES  IN  MILLILITERS  PER  GRAM  OF  MIXTURES 

OF  CARBON  TETRACHLORIDE  AND  BENZENE,  FROM  THORPE 

AND  RODGER 


Per  cen 

t  benzene  h 

y  weight 

0 

22.37 

43.79 

67.71 

100 

Per  cen 

t  benzene  b 

y  volume 

0 

34.30 

58.54 

79.17 

100 

0 

0.6127 

0.7238 
0.7242 

0.8304 
0.8309 

0.9497 
0.9501 

1.1109 

Observed 
Calculated 

10 

0.6202 

0.7326 
0.7329 

0.8405 
0.8412 

0.9610 
0.9614 

1.1242 

Observed 
Calculated 

20 

0.6278 

0.7415 
0.7418 

0.8508 
0.8511 

0.9728 
0.9730 

1  .  1377 

Observed 
Calculated 

30 

0.6355 

0.7508 
0.7509 

0.8613 
0.8615 

0.9846 
0.9849 

1.1514 

Observed 
Calculated 

40 

0.6435 

0.7602 
0.7604 

0.8724 
0.8723 

0.9971 
0.9973 

1.1661 

Observed 
Calculated 

50 

0.6518 

0.7700 
0.7702 

0.8836 
0.8836 

.0099 
.0103 

1.1812 

Observed 
Calculated 

60 

0  .  6604 

0.7801 
0.7803 

0.8951 
0.8952 

.0231 
.0234 

1  .  1966 

Observed 
Calculated 

70 

0.6(594 

0.7907 
0.7909 

0.9071 
0.9072 

.0369 
.0371 

1.2124 

Observed 
Calculated 

THE  FLUIDITY  OF  SOLUTIONS 


167 


TABLE  XLII. — THE  FLUIDITIES  OF  MIXTURES  OF  CARBON  TETRACHLORIDE 
AND  BENZENE  FROM  THORPE  AND  RODGER*  AND  FROM  LiNEBARGER2 


Temperature 

Per  cent  benzene  by  weight  (lOOn) 

0 

22.37 

43.79 

67.71 

100 

Per  cent  benzene  by  volume  (1006) 

0 

34.30 

58.54 

79.17 

100 

0 

74.1 
'72!  6 

83.6 
84.3 
86.7 
82.2 

91.9 
92.7 
95.6 
90.4 

100.6 
100.9 
103.  2 
99.  1 

110.8 
108:9 

Fluidity  observed 
0  calculated,  Eq.  (61) 
tp  calculated,  Eq.  (25) 
Fluidity  calculated  Eq.  (54) 

10 

88.2 
'88.4 

100.0 
100.6 
103.0 
99.9 

110.1 
110.6 
113.5 
110.0 

120.2 
120.2 
122.5 
119.9 

131.5 
isiis 

Observed 
<t>  calculated,  Eq.  (61) 
<p  calculated.  Eq.  (25) 
Calculated  Eq.  (56) 

20 

103.2 

ioiii 

117.6 
118.3 
120.7 
117.9 

128.9 
130.0 
133.0 
130.0 

141.4 
141.2 
143.5 
141.7 

154.1 
155:  1 

Observed 
0  calculated,  Eq.  (61) 
<p  calculated,  Eq.  (25) 
Calculated  Eq.  (56) 

30 

118.9 
126!  6 

136.2 
136.8 
139.2 
136.7 

149.0 
150.5 
153.5 
150.3 

163.4 
163.4 
165.7 
163.5 

178.0 
178:6 

Observed 
<£  calculated,  Eq.  (61) 
V  calculated  Eq.  (25) 
Calculated  Eq.  (56) 

40 

135.5 
137:5 

156.0 
156.2 
158.7 
155.7 

171.5 
172.0 
175.1 
171.8 

186.6 
186.6 
189.0 
186.3 

203.1 
263:9 

Observed 
<t>  calculated,  Eq.  (61) 
<p  calculated,  Eq.  (25) 
Calculated  Eq.  ,56) 

50 

153.0 
154!  9 

176.7 
176.5 
179.0 
175.5 

194.9 
194.2 
197.4 
193.5 

211.4 
210.6 
213.0 
210.3 

228.8 
229:9 

Observed 
<f>  calculated,  Eq.  (61) 
<p  calculated,  Eq.  (25) 
Calculated  Eq.  (56) 

60 

171.5 
i73!6 

198.8 
198.0 
200.5 
195.9 

219.31   237.0 
217.8    236.1 
221.0    238.5 
218.81  234.3 

256.1 
256:4 

Observed 
<t>  calculated,  Eq.  (61) 
<p  calculated,  Eq.  (25) 
Calculated  Eq.  (56) 

70 

191.0 
192:6 

243.31   263.9 
242.81   262.9 
246.01   265.4 
239.0!   260.1 

284.9 
283:6 

Observed 
<t>  calculated,  Eq.  (61) 
<p  calculated,  Eq.  (25) 
Calculated  Eq.  (56) 

Value  of  k... 
Value  of  w  

2,105' 
0.  5782" 

2,019    1,937    1,845  i   1  ,721'    | 
0.68310.78370.8960    1.0476. 

!              1 

Temperature 

Per  cent  benzene  by  weight 

0 

13.73    40.78    58.60 

100 

-; 

25 

113.2 
111.0 

123.8 
121.8 

141.6 
137.2 

151.5 
147.1 

166.9 
166.7 

Linebarger  observed 
Thorpe  and  Rodger  observed 

1  J.  Chem.  Soc.  (London),  71,  364  (1897). 

2  Am.  J.  Set.  (4).  11,  331  (1896). 

3  Batschinski,    Zeitschr.  f.   physik.   Chem.,  84,   643   (1913).     Calculated  on  the  basis  of 
Young's  specific  gravities. 


168 


FLUIDITY  AND  PLASTICITY 


which  have  been  studied  lately.  Kendall  and  Wright  and  many 
others  have  done  valuable  service;  they  have  chosen  inert  liquids 
whose  individual  fluidities  are  widely  separated,  hence  these 
mixtures  are  suited  to  give  a  crucial  test  of  the  mixture  formulas. 
Delbert  F.  Brown  has  studied  this  data  to  determine  (1)  whether 
the  volume  difference  Av  is  greatest  when  the  specific  gravities 
of  the  components  are  most  widely  different,  (2)  whether  the 
fluidity  difference  A<p  is  greatest  in  the  same  mixtures  in  which  the 
volume  difference  is  greatest,  (3)  whether  those  pairs  of  liquids 
showing  the  greatest  volume  difference  also  show  the  greatest 
fluidity  divergence,  and  (4)  whether  the  fluidities  of  the  mixtures 
can  be  calculated  from  the  fluidities  and  volumes  of  the  compo- 
nents. Table  XLIII  gives  a  summary  of  part  of  his  results. 
The  table  is  so  arranged  that  the  differences  between  the  specific 
volumes  of  the  components,  column  2,  are  in  the  order  of  increas- 

TABLE   XLIII. — FLUIDITY   AND    VOLUME    RELATIONS  IN  CERTAIN  IDEAL 
MIXTURES   (USING   DATA   OF   KENDALL   AND   WRIGHT) 


Specific  vol- 

Components 

ume  differ- 
ence of  com- 

Specific vol- 
ume differ- 

Fluidity dif- 
ferencc  A^r 

Average  de- 
viation in 

ponents 

ence  At, 

per  cent 

PJ    —  7)1 

Ethyl      benzoate      and      benzyl 

benzoate  

0.0594 

0.0010 

5.04 

1.0 

Phenetol  and  diphenyl  ether  

0.1056 

0.0028 

5.58 

0.56 

Ethyl  acetate  and  ethyl  benzoate 

0.1616 

0.0062 

25.9 

1.2 

Ethyl  acetate  and  benzyl  benzoate1        0.  2183 

0.0118 

55.05 

9.2 

Diethyl  ether  and  phenetol  

0.3610 

0.0268 

60.3 

1.2 

Diethyl  ether  and  diphenyl  ether 

0.4666 

0.0458 

110.5 

3.8 

ing  magnitude.  The  third  column  shows  that  the  sag  in  the 
volume-volume  concentration  curve  follows  exactly  this  order 
of  increase,  and  column  4  shows  that  the  fluidity  divergence  A«p 
follows  the  same  order  of  increase.  Moreover,  the  maximum 
divergence  in  both  the  volumes  and  the  fluidities  occurs  in  the 
same  mixture  in  every  case,  except  that  of  diethyl  ether  and 
diphenyl  ether,  although  it  is  not  possible  to  bring  this  out  in 
the  table.  The  last  column  shows  the  average  deviation  of  the 
values  of  the  fluidity,  as  calculated  by  Brown  from  the  data  of 
Kendall  and  Wright  by  the  formula  (61).  In  only  two  cases  is 


THE  FLUIDITY  OF  SOLUTIONS  169 

this  deviation  much  over  1  per  cent.  Brown  found  that  the 
deviation  is  usually  larger  in  mixtures  which  contain  an  ester  as 
one  or  both  of  the  components.  This,  however,  is  not  shown  by 
this  table  very  well,  but  if  the  conclusion  is  correct,  the  deviation 
would  be  explained  by  the  chemical  character  of  the  components. 
This  brings  us  to  the  consideration  of  the  non-ideal  types  of 
mixtures. 

The  reader  will  perhaps  ask  whether  the  fluidities  of  ideal 
mixtures  would  be  additive  if  plotted  against  weight  concentra- 
tions. The  curves  for  carbon  tetrachloride  and  benzene  have 
been  published,1  using  both  volume  and  weight  concentrations. 
Using  volume  concentrations  the  curves  are  slightly  sagged  as 
already  pointed  out,  but  using  weight  concentrations  they  show 
marked  negative  curvature  particularly  at  the  higher  tempera- 
tures. The  very  slight  contraction  of  carbon  tetrachloride  and 
benzene  on  mixing  in  no  way  accounts  for  this  negative 
curvature. 

II.  NEGATIVE  CURVATURE  AND  DISSOCIATION  BY  DILUTION 

We  will  now  consider  a  pair  of  substances  which  expand  on 
mixing,  using  the  data  of  Thorpe  and  Rodger  for  methyl  iodide 
and  carbon  disulphide.  The  curvature  of  the  fluidity-volume  con- 
centration curves  is  negative  and  greatest  at  the  lowest  tempera- 
tures. This  is  in  accordance  with  the  view  that  the  components 
are  less  associated  at  the  higher  temperatures  and  therefore  can 
show  less  dissociation  on  mixing. 

The  expansion  on  mixing  amounts  to  as  much  as  0.2  per  cent 
of  the  volume,  as  may  be  seen  by  comparing  the  observed  specific 
volumes  with  those  calculated  by  the  admixture  rule,  Table 
XLV.  The  fluidities  are  given  in  Table  XLIV  and  it  is  seen  that 
Batschinski's  Law  applies  to  each  mixture,  but  the  values  of  the 
limiting  specific  volumes  w  cannot  be  calculated  by  the  admixture 
rule  as  in  the  normal  mixture.  The  actual  limiting  volume  is 
some  2  per  cent  less  than  the  calculated  value,  presumably  due 
to  the  dissociation.  The  values  of  k,  which  measure  the  slope 

-  of  the  fluidity-specific  volume  curves  are  very  much  less 


Zeitschr.  f.  phijsik.  Chem.,  83,  657  (1913). 


170 


FLUIDITY  AND  PLASTICITY 


than  the  calculated  values.     This  is  also  in  marked  contrast  to 
the  case  of  carbon  tetrachloride  and  benzene. 

TABLE  XLIV. — THE  FLUIDITIES  OP  MIXTURES  OP  METHYL  IODIDE  AND 
CARBON  BISULPHIDE  (FROM  THORPE  AND  RODGER) 


Temper- 
ature 

Per  cent  carbon  disulphide  by  weight 

0 

21.60 

38.81 

48.11 

68.81 

82.39 

100 

Per   cent   carbon   disulphide   by   volume 

0 

32.22 

53.3, 

62.61 

79.94 

89.42 

100 

0 

168.3 
168.6 

193.1 
193.0 
192.3 

207.5 
207.5 
207.6 

213.2 
212.5 
213.3 

222.7 
223.1 
223.0 

228.3 

228.7 
228.4 

232.8 
233.1 

Observed 
Calculated,  Batschinski 
Calculated,  Gibson 

10 

186.6 
186.2 

211.4 
211.3 
211.1 

225.7 
225.6 
225.7 

225.9 
230.5 
230.0 

242.1 
241.6 
242.0 

248.1 
248.0 
248.0 

253.0 
252.1 

Observed 
Calculated,  Batschinski 
Calculated,  Gibson 

20 

205.3 
205.3 

230.9 
230.4 
229.2 

243.9 
243.8 
244.0 

248.1 
248.5 
248.2 

261.8 
260.6 
262.0 

268.1 
268.0 
268.0 

272.5 
271.9 

Observed 
Calculated,  Batschinski 
Calculated,  Gibson 

30 

224.8 
224.7 

250.6 
250.2 
250.8 

263.2 
262.8 
263.2 

267.4 
267.0 
268.0 

280.1 
280.0 
281.0 

288.2 
288.2 
288.0 

293.5 
292.2 

Observed 
Calculated,  Batschinski 
Calculated,  Gibson 

40 

244.7 
244.8 

271.0 
270.2 
270.8 

282.5 
282.3 
282.6 

285.7 
286.2 
286.0 

299.4 
300.1 
300.0 

309.6 
309.8 
309.5 

314.0 
313.4 

Observed 
Calculated,  Batschinski 
Calculated,  Gibson 

Value  of  « 
from   mix- 
tures   
from     sol- 
vents   

0.3808' 

0.4405 
0.4420 

0.4855 
0.4908 

0.5096 
0.5171 

0.5722 
0.5758 

0.6161 
0.6143 

0.6642 

Value  of  k 
from   mix- 
tures. .  .  . 
from     sol- 
vents .... 

3,527' 

3,040 
3,224 

2,633 
2,982 

2,465 
2,852 

2,341 
2,561 

2,322 
2,370 

2,123 

1  Using  the  densities  of  Thorpe  and  Rodger  we  have  recalculated  these  constants,  instead 
of  taking  the  values  of  Batschinski. 

When  there  was  a  volume  change  on  mixing,  Gibson  assumed 
that  the  specific  volumes  v\  and  v*  were  not  the  same  as  for  the 


THE  FLUIDITY  OF  SOLUTIONS 


171 


components.  He  assumed  that  the  free  volume  per  unit  of 
limiting  volume  was  the  same  for  each  kind  of  molecule,  so  that 
from  the  equations 


and 


the  values  of  Vi  and  vz  could  be  calculated.  He  then  calculated 
the  fluidities  of  the  mixture  by  means  of  the  simple  additive 
fluidity  formula  (25).  That  the  values  calculated  by  Gibson 
agree  well  with  the  observed  is  shown  in  the  third  line  of  fluidities 
for  each  temperature  given  in  Table  XLIV. 


TABLE  XLV. — THE   SPECIFIC  VOLUMES  IN  MILLILITERS   PER   GRAM   OF 

MIXTURES  OF  METHYL  IODIDE  AND  CARBON  DISULPPIDE  (FROM 

THORPE  AND  RODGER) 


Temper- 
ature 

Per   cent    carbon    disulphide   by   volume 

0 

33.22 

53.39 

62.61 

79.94 

89.42 

100 

0 

0.4285 

0.5040 
0.5031 

0.5643 
0.5624 

0.5959 
0  .  5945 

0.6675 
0  .  6659 

0.7146 
0.7128 

0.7740 

Observed 
Calculated 

10 

0.4336 

0.5100 

0.5712 

0.6031 

0.6754 

0.7229 

0.7830 

Observed 

20 

0.4390 

0.5163 
0.5152 

0.5781 
0.5759 

0.6104 
0.6087 

0.6835 
0.6817 

0.7315 
0.7297 

0.7923 

Observed 
Calculated 

30 

0.4445 

0.5228 

0.5853 

0  6179 

0.6918 

0.7412 

0.8018 

Observed 

40 

0.4502 

0.5295 
0.5282 

0.5927 
0.5904 

0.6257 
0.6242 

0.7004 
0.6987 

0.7495 
0.7475 

0.8118 

Observed 
Calculated 

We  have  come  now  to  the  case  where  there  is  chemical  com- 
bination on  mixing.  There  is  generally  a  decrease  in  volume 
and  the  specific  volume-weight  concentration  curve,  curve  II, 
Fig.  60,  is  sagged  as  well  as  curve  V,  Fig.  61,  representing  the 
specific  volume-volume  concentration  curve.  Since  new  sub- 
stances are  formed,  no  method  given  thus  far  can  be  depended 
upon  for  calculating  the  fluidity-volume  concentration  curve. 


172  FLUIDITY  AND  PLASTICITY 

III.  POSITIVE  CURVATION  AND  CHEMICAL  COMBINATION 

Before  considering  the  meaning  of  positive  curvature  in  detail, 
it  is  necessary  to  emphasize  the  fact  that  a  minimum  in  the  fluid- 
ity-volume concentration  curve  is  not  necessary  to  indicate  that 
chemical  combination  is  taking  place  and  when  a  minimum  does 
occur,  its  location,  according  to  numerous  investigators,  notably 
Findlay  (1909)  and  Denison  (1913),  does  not  correspond  to  the 
exact  composition  of  the  compound  formed.  This  is  proved,  if 
proof  be  needed,  by  the  fact  that  the  minimum  usually  changes 
with  the  temperature  and  may  disappear  altogether.  The  ques- 
tion then  is,  assuming  that  a  chemical  combination  is  formed  by 
the  mixing  of  the  two  components  of  a  binary  mixture,  how  can  the 
data  be  used  to  show  what  this  compound  is?  To  answer  this, 
we  will  present  three  cases  of  increasing  complexity,  in  all  of 
which  there  is  the  same  amount  of  chemical  combination,  it  being 
assumed  that  in  the  feeble  combination  with  which  we  are  dealing 
the  two  components  A  and  B  are  always  in  equilibrium  with  a 
small  amount  of  the  compound  C  so  that 

A  +  B+±C 

Case  I. — The  fluidity-temperature  curves  of  two  closely  related 
substances  are  represented  by  the  curves  A  and  B  in  Fig.  63a. 
If  there  were  no  combination  between  the  components  on  mixing, 
the  curve  for  the  50  per  cent  mixture  would  lie  half-way  between 
the  curves  A  and  B  (dotted).  Let  it  be  assumed  that  this  mix- 
ture does  show  the  maximum  amount  of  combination  and  that 
the  curve  is  thereby  lowered  to  0.5.6.  Using  the  data  plotted  in 
Fig.  63a  it  becomes  possible  to  plot  the  fluidity-volume  concen- 
tration curves  for  the  various  temperatures  t\,  tz,  tz,  etc.,  as  shown 
in  Fig.  636.  In  this  case  there  is  a  well-defined  minimum  in  the 
fluidity-volume  concentration  curve  in  the  50  per  cent  mixture 
and  the  deviation  of  the  curves  from  the  normal  (dotted)  curves 
is  constant  in  amount. 

Case  II. — Let  us  now  assume  that  we  are  dealing  with  two 
substances  whose  fluidities  are  widely  different,  although  they 
still  run  parallel  to  each  other.  With  the  same  amount  of  combi- 
nation as  before,  the  curve  0.5B  falls  between  the  curves  A  and  B, 
Fig.  64a.  As  a  result  the  fluidity-volume  concentration  curves, 
Fig.  646,  no  longer  exhibit  a  minimum  although,  by  assumption, 


THE  FLUIDITY  OF  SOLUTIONS 


173 


the  hydration  is  the  same  as  before  both  in  relative  composition 
and  amount.     However,  it  is  clear  that  the  deviation  of  the  fluidity 


Temp.  A        %  JB 

a  b 

FIG.  63. — Diagram  to  illustrate  the  fact  that  when  two  substances  A  and  B 
of  similar  fluidity  are  mixed,  the  formation  of  a  solvate  produces  a  minimum  in 
the  fluidity-concentration  curves. 


i*  o 

FIG.  64. — Diagram  illustrating  how  when  two  substances  A  and  B  are  mixed 
whose  fluidities  are  very  different,  the  formation  of  a  solvate  produces  no  mini- 
mum in  the  fluidity-concentration  curve. 

volume  concentration  curves  from  the  linear  curves,  which  would 
be  expected  were  there  no  combination,  and  as  indicated  in  the 
figures  by  the  distance  MN,  is  the  same  as  in  the  preceding  case. 


174 


FLUIDITY  AND  PLASTICITY 


Case  III. — In  the  usual  case  in  practice,  the  fluidity-tempera- 
ture curves  are  not  parallel,  so  that  the  fluidities  may  be  identical 
at  one  temperature  but  very  different  at  another.  We  then 
obtain  a  series  of  curves  as  shown  in  Fig.  65a  and  656.  At  low 
temperatures  there  is  a  good  minimum  in  the  fluidity-volume 
concentration  curves,  but  it  gradually  shifts  to  the  right  as  the 
temperature  is  raised,  until  at  the  highest  temperatures  it  dis- 
appears altogether.  It  is  manifestly  erroneous  to  assume  that 
the  composition  of  the  hydrate  changes  on  this  account.  On  the 
other  hand,  the  deviation  from  the  expected  linear  curves  as 


FIG.  65. — Diagram  illustrating  how  the  minimum  in  the  fluidity-concentra- 
tion curve  may  shift  with  the  temperature.  The  maximum  deviation  from  the 
linear  curve  is  the  significant  quantity.  This  quantity  does  not  vary  with  the 
temperature  and  it  indicates  the  composition  of  the  solvate. 


measured  vertically  is  everywhere  the  same  as  in  the  simpler 
cases.  In  practice,  the  hydration  is  generally  less  at  the  higher 
temperatures  so  that  the  deviation  should  grow  less  as  the  tem- 
perature is  raised,  but  the  cases  already  given  are  sufficient  to 
show  that  the  deviation  of  the  observed  fluidity-volume  concentra- 
tion curve  from  the  linear  curve,  which  would  be  expected  were 
there  no  combination  between  the  components  of  the  solution, 
can  alone  furnish  trustworthy  information. 

Were  the  components  of  the  mixture  non-associated,  it  seems 
possible  to  calculate  not  only  the  composition  of  the  solvate 
formed  but  also  the  percentage  of  it  existing  in  the  solution. 
But  substances  which  form  feeble  combinations  on  mixing  are 
usually  themselves  associated,  and  it  is  quite  likely  that  this 


THE  FLUIDITY  OF  SOLUTIONS  175 

association  is  altered  in  the  mixture,  so  that  the  result  is  consid- 
erably complicated  thereby.  We  have,  however,  a  fairly  simple 
case  in  mixtures  of  ether  and  chloroform  studied  by  Thorpe  and 
Rodger.  Chloroform,  like  carbon  tetrachloride,  is  probably 
slightly  associated  but  ether  may  be  regarded  as  unassociated. 

So  far  as  can  be  learned  from  their  measurements  the  maximum 
contraction  on  mixing  occurs  in  a  mixture  containing  less  than 
40  per  cent  of  ether  and  perhaps  less  than  39  per  cent;  the  maxi- 
mum deviation  of  the  fluidity-volume  concentration  curve  from  the 
linear  curve  occurs  in  the  58  volume  per  cent  mixture  +  3  per  cent. 
This  corresponds  to  39.8  per  cent  by  weight.  A  mixture  corre- 
sponding to  the  formula  C-iHioO.CHClg  contains  38.30  per  cent 
ether  by  weight.  Guthrie  has  noted  that  heat  is  evolved  on 
mixing  and  that  it  is  a  maximum  when  the  components  are  in 
molecular  proportions.  The  vapor-pressure,  refractive  index 
and  the  freezing-point  curves  all  give  evidence  of  the  formation 
of  a  compound  C4Hi0O.CHCl3. 

In  the  mixture  containing  56.26  volume  per  cent  of  ether,  or 
one  molecule  of  ether  to  one  of  chloroform,  we  will  now  calculate 
the  percentage  combined.  From  the  atomic  constants  already 
given,  p.  126,  it  appears  that  the  compounds  C^ioO.CHCls 
should  have  a  fluidity  of  200  at  the  absolute  temperature  of  538.6°. 
But  actually  a  mixture  of  this  composition  has  a  fluidity  of  200 
at  282.9°  absolute  (9.9°C).  Pure  ether  and  pure  chloroform 
have  fluidities  of  200  at  216.5°  and  305.3°  absolute  respectively, 
so  that  if  the  mixture  were  wholly  uncombined,  the  absolute 
temperature  necessary  for  a  fluidity  of  200  would  be  216.5  X 
0.5626  +  305.6  X  0.4374  =  255.4°.  Letting  x  represent  the 
fraction  of  the  volume  of  the  mixture  which  is  combined,  we  ob- 
tain the  equation 

538.6*  +  255.4(1  -  x)  =  282.9 

and  x  =  0.0971.  Since  at  this  temperature  (9.9°C)  less  than  10 
per  cent  of  the  volume  of  the  mixture  is  actually  in  combination, 
it  seems  reasonable  to  assume  that  a  dynamic  equilibrium  exists 
between  the  combined  and  the  uncombined  portions.  If  the 
Mass  Law  holds,  we  have 

[C4H100]  [CHC13] 


[C4H100  .  CHC13] 


=  K 


176  FLUIDITY  AND  PLASTICITY 

where  the  concentrations  are  molecular  and  not  volume  concen- 
trations. 

In  the  above  equimolecular  mixture,  if  we  let  y  represent  the 
number  of  milliliters  of  ether  which  are  combined  in  every  100 
ml  of  mixture,  the  volume  of  the  chloroform  combined  will  be 
0.7366  X  119.36y       .  ___„ 

74.08X1.526      =  °'7777y 

where  the  specific  gravities  of  ether  pe  and  chloroform  pc  are 
taken  as  0.7366  and  1.526  respectively  and  their  molecular 
weights,  me  and  mc,  74.08  and  119.36.  Since  the  sum  of  the  two 
volumes  y  +  0.7777y  is  9.71,  the  volume  of  the  ether  combined 
is  5.46  ml  and  the  volume  of  the  chloroform  is  4.25  ml. 

Substituting  the  molecular  concentrations  in  the  above  formula 
[(56.26  -  5.46)Pe/me][(43.74  -  4.25)Pc/mc] 

9.71p/(me+mc) 

where  p  is  the  density  of  the  compound  calculated  by  averages 
to  be  1.082. 

With  this  value  of  K,  it  is  possible  to  calculate  the  absolute 
temperature  corresponding  to  a  fluidity  of  200  for  any  mixture 
on  the  assumption  that  only  one  compound  is  formed  and  that 
the  Law  of  Mass  Action  is  obeyed.  Thus  for  any  mixture  if  a 
is  the  volume  percentage  of  ether  and  z  is  the  fraction  of  the 
ether  which  is  combined, 

(1    —   &)PC  &(• 


me  L       mc  me 

me 

For  the  28.21  volume  per  cent  ether  mixture,  z  =  0.157.  The 
volume  of  ether  in  100  ml  which  is  combined  is  az  =  4.43  ml, 
and  the  volume  of  chloroform  combined  is  0.7777  X  4.43  = 
3.44  ml.  Hence  the  calculated  absolute  temperature  correspond- 
ing to  a  fluidity  of  200  is 

0.2378  X  216.5  +  0.6835  X  305.3  +  0.0787  X  538.6  =  302.5° 
which  is  in  fair  agreement  with  the  value  read  from  the  curve 
of  297.4°. 

ETHYL  ALCOHOL  AND  WATER  MIXTURES 

We  will  now  take  up  a  case  in  which  the  components  of  the 
mixture  are  highly  associated.  Ethyl  alcohol  and  water  are  a 


THE  FLUIDITY  OF  SOLUTIONS 


177 


particularly  good  example  as  there  is  a  very  strongly  pronounced 
minimum  in  the  fluidity  curves.  The  greatest  deviation  from 
the  linear  is  in  a  mixture  corresponding  to  the  formula  C2H6O.- 
4H2O,  containing  44.79  per  cent  alcohol  by  volume.  To  obtain 
a  fluidity  of  200,  ethyl  alcohol  requires  an  absolute  temperature 
of  343.6°  and  water  a  temperature  of  328.9°,  so  if  there  were  no 
chemical  change  on  mixing  we  should  expect  a  temperature 
corresponding  to  the  fluidity  of  200  in  the  44.79  volume  per  cent 


150 
140 
130 
120 
110 
!?100 

'•5  90 

I » 

70 
60 
50 
40 


20         40         60         80    •    100 


FIG.  66. — Mixing  solutions  of  ethyl  alcohol  in  water  (corresponding  to  the 
composition  C2HeO.3H2O)  with  solutions  of  acetic  acid  in  water  (corresponding 
to  the  composition  C2H4O2.H2O)  brings  about  an  increase  in  fluidity. 


mixture  of  0.4479  X  343.6  X  0.5521  X  328.9  =  335.5°.  On  the 
other  hand,  from  the  constants  already  given,  the  temperature 
required  to  give  the  pure  hydrate  C2H6O.4H2O  a  fluidity  of  200 
would  be  14  X  59.2  +  5  X  24.2  -  2  X  95.7  =  758.4°.  But  the 
observed  absolute  temperature  at  which  the  44.79  volume  per 
cent  mixture  has  a  fluidity  of  200  is  362.3°.  Hence,  if  we  let 
x  represent  the  fraction  by  volume  of  the  mixture  combined  as 
C2H60.4H20,  the  rest  remaining  unchanged,  we  have 

335.5  (1  -  x)  +  758.4z  =  362.3 
and  x  =  0.0634. 

12 


178  FLUIDITY  AND  PLASTICITY 

That  ethyl  alcohol  and  water  are  less  than  7  per  cent  combined 
is  surprising  in  view  of  the  higher  amount  of  combination  in 
chloroform  and  ether,  but  the  temperature  of  comparison  is 
very  much  higher,  in  the  case  of  water  and  alcohol  being  89°C. 
But  there  is  another  important  disturbing  factor  which  must  be 
considered,  in  that  water  and  alcohol  are  both  highly  associated, 
2.31  and  1.83  respectively,  so  that  when  the  two  are  mixed  there 
is  almost  certainly  dissociation. 

That  dissociation  does  occur  can  be  proved  as  follows:  We 
have  seen  that  when  ethyl  alcohol  and  water  are  mixed  there  is 
a  lowering  of  the  fluidity.  There  is  also  a  pronounced  lowering 
of  the  fluidity  when  acetic  acid  and  water  are  mixed.  There  is 
furthermore  a  lowering  of  the  fluidity  when  acetic  acid  is  mixed 
with  ethyl  alcohol.  Yet  when  acetic  acid  solution  (C2H4O2.H2O) 
is  mixed  with  ethyl  alcohol  solution  (C2H6O.3H20),  having 
practically  the  same  fluidity  of  43  absolute  units  at  25°C,  there 
is  a  very  noticeable  increase  in  the  fluidity  as  seen  in  Fig.  66  from 
the  paper  by  Bingham,  White,  Thomas  and  Cadwell  (1913). 

IV.     INFLECTION  CURVES 

The  discussion  of  simultaneous  dissociation  and  chemical 
combination  brings  us  naturally  to  the  consideration  of  the 
fourth  type  of  fluidity-volume  concentration  curves.  There 
are  several  pairs  of  non-aqueous  mixtures  which  fall  into  this 
class,  such  as  ethyl  alcohol  and  benzene  discovered  by  Dunstan 
(1904);  but  by  far  more  important  are  certain  aqueous  solutions  of 
electrolytes,  notably  the  salts  of  potassium,  rubidium,  caesium 
and  ammonium.  That  potassium  nitrate  added  to  water  lowers 
the  time  of  flow  was  discovered  by  Poiseuille  (1847)  although 
priority  is  usually  attributed  to  Hiibener  (1873).  The  list  of  those 
substances  which  lower  the  viscosity  of  water  has  been  added  to 
by  Sprung  (1875),  Slotte  (1883)  and  many  others  and  is  given  in 
Table  XL VI.  The  phenomenon  has  been  often  referred  to  as 
"negative  viscosity,"  but  since  viscosity  is  a  result  of  friction, 
which  is  never  negative  in  fact,  the  use  of  the  term  is  not  happy. 
The  term  "negative  curvature,"  d2  <p/db*<  0,  where  b  is  the 
volume  concentration,  is  not  open  to  similar  objection  when  dis- 
cussing the  fluidity- volume  concentration  curves  of  these  solutions. 


THE  FLUIDITY  OF  SOLUTIONS 


179 


Apparently  all  of  those  aqueous  solutions  which  exhibit  negative 
curvature  fall  into  the  class  of  mixtures  showing  inflection  curves. 

TABLE  XLVI. — SUBSTANCES    WHICH  APPEAR  TO  EXHIBIT  THE  SO-CALLED 
"NEGATIVE  VISCOSITY"  IN  AQUEOUS  SOLUTION 


Substance 


Observer 


Bromic  acid 

Hydrobromic  acid 

Hydrocyanic  acid 

Hydriodic  acid 

Hydrosulfuric  acid 

Nitric  acid 

Ammonium  acetate 

Ammonium  bromide 

Ammonium  chloride 

Ammonium  chromate 

Ammonium  iodide 

Ammonium  nitrate 

Ammonium  thiocyanate 

Caesium  chloride 

Caesium  nitrate 

Ferrous  iodide 

Mercuric  chloride 

Mercuric  cyanide 

Potassium  bromide 

Potassium  chlorate 

Potassium  chloride 

Potassium  cyanide 

Potassium  ferricyanide 

Potassium  iodide 

Potassium  nitrate 

Potassium  thiocyanate 

Rubidium  bromide 

Rubidium  chloride 

Silver  nitrate 

Sodium  iodide 

Tetramethylammonium  iodide 

Thallium  nitrate 

Urea... 


Poiseuille  (1847) 

Poiseuille  (1847) 

PoiseuiUe  (1847) 

Poiseuille  (1847 

Poiseuille  (1847 

Poiseuille  1847) 

Poiseuille  (1847) 

Sprung  (1876) 

Poiseuille  (1847) 

Schlie  (1869) 

Hubener  1873),  Wagner  (1890) 

Sprung  (1876),  Gorke  (1905),  Walden  (1906) 

Sprung  (1876) 

Schottner  (1878) 

Bruckner  (1891) 

Poiseuille  (1847) 

Poiseuille  (1847) 

Slotte  (1883),  Ranken  and  Taylor  (1906) 

Poiseuille  (1847) 

Poiseuille  (1847) 

Poiseuille  (1847) 

Poiseuille  (1847) 

Hubener  (1873),  Kanitz  1897) 

Poiseuille  (1847) 

Poiseuille  (1847) 

Sprung  (1876),  Gorke  (1905) 

Davis,  Hughes  and  Jones  (1913) 

Wagner  (1890) 

Poiseuille  (1847) 

Poiseuille  (1847) 

Schlie  (1869) 

Schottner  (1878) 

Mtitzel  (1891) 


Most  workers  have  confined  their  attention  to  dilute  solutions 
and  they  have  studied  viscosity  relations  almost  exclusively, 
so  that  the  positive  curvature  in  concentrated  solutions  has 


180 


FLUIDITY  AND  PLASTICITY 


350' 


FIG.  67. — Fluidity-volume  concentration  curves  for  aqueous  solutions  of  ammo- 
nium nitrate,  showing  both  positive  and  negative  curvature. 


THE  FLUIDITY  OF  SOLUTIONS 


181 


remained  undiscovered.  That  positive  curvature  is  general  even 
when  it  is  quite  unsuspected  in  dilute  solutions  is  proved  by 
all  of  the  data  available,  as  may  be  seen  by  inspection  of  Figs. 
67,  68,  69  and  70.  The  negative  curvature  is  in  each  case  most 


150 


\ 


\\. 


\ 


v\ 


175 


125 


0     10    20     JO    40    50    60     10     80    90    100 
Fluidity  of  Solutions,  of  Urea 

FIG.  68. 


0     10    20     30    40    50    60    10    80    90   100 
Fluidity  of  Solutions  of  Silver  Nitrate 

FIG.  69. 


pronounced  at  the  lowest  temperatures  and  in  solutions  contain- 
ing not  over  20  per  cent  of  the  salt  by  volume.  The  negative 
curvature  disappears  in  concentrated  solutions  and  at  high 
temperatures.  If  the  negative  curvature  is  strongly  marked, 
as  with  ammonium  nitrate  or  potassium  chloride,  the  positive 
curvature  is  unimportant,  but  when  the  negative  curvature 


182 


FLUIDITY  AND  PLASTICITY 


is  weak,  as  with  potassium  iodide,  silver  nitrate  and  urea,  the 
positive  curvature  quickly  shows  itself. 

The  first  important  attempt  to  explain  the  lowering  of  the 
viscosity  of  water  was  made  by  Arrhenius  (1887),  who  thought 
that  it  might  be  due  to  electrolytic  dissociation.  Wagner  and 


220 
200 
180 

160 
140 
120 
100 
80 
60 
40 
20 

°< 

r^ 

V 

FLUIDITIES  OFSOLUTIONS 
OF  POTASSIUM  SALTS 

\ 

KN 

oC 
^  P 

nlori 
romk 

jdide 

jle 
le 

u^ 

( 

& 

Dl< 

k 

Ss 

^ 

\ 

X 

\ 

XN 

itrah 

•a 

^ 

^ 

\ 

v 

«£\ 

—  o—  » 

^"XX 

^\ 

\ 

\ 

s 

&< 

N 

N 

-«~. 

s 

s> 

\ 

\ 

^f^ 

g 

N 

vx 
\ 
\ 

\ 

. 

?<?--. 

•>- 

\    ^ 

V   ^>. 

\ 

Xx^ 

\ 

*^ 

•*^ 

^^ 

N 
^^ 

\ 
X    \ 

N  \ 

"^ 

-<:: 

^ 

10        20       30       40        50      60      10        80       90      IOC 
Per  Cent 

FIG.  70. — Fluidities  of  potassium  halide  solutions  in  water  at  various  tempera- 
tures. The  curves  show  negative  curvature  which  is  most  marked  for  the 
chloride,  and  at  low  temperatures  and  at  low  volume  concentrations  of  the  salt. 
At  high  concentrations  or  at  high  temperatures  all  of  these  solutions  may  show 
positive  curvature,  but  the  nitrate  and  iodide  most  readily.  (After  Gorke.) 

Miihlenbein  (1903),  however,  showed  that  the  dissociation 
hypothesis  was  by  itself  insufficient  as  an  explanation  since  salts 
like  NaNO3  and  K2SO4  are  highly  ionized  and  yet  do  not  show 
negative  curvature  as  does  KNO3.  Now  that  it  appears  that 
urea  and  mercuric  chloride  solutions  both  show  negative  curva- 
ture, it  would  seem  probable  that  electrolytic  dissociation  is 
not  necessary  for  the  phenomenon.  Since  these  substances  in 
solution  are  practically  unionized. 


THE  FLUIDITY  OF  SOLUTIONS  183 

Jones  and  Veazey  (1907)  observed  that  potassium,  rubidium 
and  caesium  are  the  elements  with  the  largest  atomic  volume  and 
they  therefore  reasoned  that  their  salts  would  also  be  relatively 
fluid.  From  what  has  preceded  we  are  prepared  to  find  relations 
between  fluidity  and  volume,  but  as  a  matter  of  fact  the  fluidity 
of  the  pure  salts  in  the  molten  condition  is  very  low.  For 
example,  Foussereau  (1885)  found  the  fluidity  of  ammonium 
nitrate  to  be  0.505  at  185°C  and  0.4037  at  162°C,  so  that  at  ordi- 
nary temperatures  the  fluidity  of  the  salt  in  the  undercooled 
condition  would  certainly  be  very  low,  probably  negligible  as 
compared  with  water.  Furthermore,  there  are  salts  which  show 
negative  curvature  but  in  which  the  metal  has  a  small  atomic 
volume  such  as  silver  nitrate,  mercuric  chloride,  and  thallium 
nitrate.  In  view  of  the  periodic  relationship  of  the  elements, 
the  same  coincidence  noted  by  Jones  and  Veazey  would  occur 
with  many  other  properties.  Finally  there  are  several  salts  of 
potassium  and  ammonium  which  have  not  been  found  to  show 
negative  curvature  hence  the  explanation  proposed  by  Jones  and 
Veazey  is  not  satisfactory. 

EXPLANATION  OF  THE  INFLECTED  CURVE 

As  to  the  reason  for  positive  curvature,  it  seems  probable 
from  what  precedes  that  it  is  due  to  combination  between 
the  solvent  and  the  solute.  That  many  of  the  salts  of  potassium, 
rubidium,  caesium  and  ammonium  exhibit  so  slight  positive 
curvature  is  due  to  their  smaller  tendency  to  form  hydrates 
than  is  usually  the  case  in  aqueous  solution.  In  contrast  with 
the  salts  of  potassium,  no  sodium  salts  show  "negative  viscosity." 
Perhaps  the  most  striking  difference  between  the  salts  of  sodium 
and  potassium,  generally  so  similar,  is  the  greater  affinity  for 
water  on  the  part  of  sodium  salts.  None  of  the  salts  which  show 
negative  curvature  crystallize  from  water  with  water  of  crys- 
tallization, and  the  few  salts  of  potassium  and  ammonium  which 
do  not  show  negative  curvature  do  exhibit  a  tendency  to  form 
hydrates.  Examples  are  potassium  carbonate,  ferrocyanide  and 
sulfate,  and  ammonium  sulfate.  It  is  true  that  hydrobromic 
acid  solutions  are  probably  hydrated,  but  according  to  the 
measurements  of  Steele,  Mclntosh,  and  Archibald  (1906)  anhy- 
drous liquid  hydrogen  bromide  has  a  high  fluidity.  The  small- 


184  FLUIDITY  AND  PLASTICITY 

ness  of  the  positive  curvature  is  then  due  to  the  small  amount  of 
hydration  which  is  well-nigh  universal  in  aqueous  solution. 

The  negative  curvature,  on  the  other  hand,  must  be  due  to 
dissociation  either  (1)  of  the  salt  or  (2)  of  the  associated  water. 
Since  the  negative  curvature  occurs  in  dilute  solution,  the 
electrolytic  dissociation  is  immediately  suggested.  If  the  fluidity 
of  the  anhydrous  salt  in  the  form  of  an  undercooled  liquid  is 
negligibly  small,  it  is  hard  to  conceive  of  how  the  dissociation  of 
the  salt  into  two,  or  at  the  most  a  few,  ions  would  increase  the 
fluidity  so  remarkably,  for  it  must  be  remembered  that  there 
must  be  a  substance  present  whose  fluidity  is  higher  than  that  of 
water.  Then,  as  already  pointed  out,  there  are  substances 
which  give  negative  curvature  which  are  very  slightly  dissociated 
into  ions,  such  as  urea. 

We  are  then  compelled  to  seek  further  in  our  explanation 
and  admit  that  water  itself  is  dissociated  by  the  presence  of  the 
salt  or  its  ions.  There  is  nothing  inherently  improbable  in  this 
since  water  is  highly  associated  (2.3  at  56°C).  The  association 
is  less  at  high  temperatures  and  in  concentrated  solutions  so 
that  under  these  conditions  negative  curvature  would  be  less 
apparent  as  we  have  already  seen  to  be  the  case.  It  is  often 
assumed  that  electrolytic  dissociation  is  brought  about  by  union 
of  simple  water  molecules  with  the  ions  of  the  salt,  but  if  the  ions 
have  low  fluidity,  the  fluidity  of  the  sohition  will  evidently  not  be 
raised  by  uniting  with  even  simple  water  molecules,  hence 
hydration  will  not  explain  the  phenomenon.  In  other  words,  the 
formation  of  larger  molecules  does  not  tend  to  raise  the  fluidity. 

Wagner  (1890)  has  measured  the  volume  of  water  required  to 
make  a  liter  of  normal  solution  of  the  chlorides  of  various  salts. 
In  the  cases  of  silver  and  thallium  the  nitrates  were  used  instead. 
Salts  like  calcium  chloride,  which  unite  strongly  with  water  to 
form  hydrates,  produce  a  contraction  on  going  into  solution,  so 
that  a  comparatively  large  volume  of  water  is  required.  But 
rubidium  and  caesium  chlorides  expand  on  going  into  solution  so 
that  the  volume  of  water  required  is  correspondingly  small.  The 
difference  between  the  volume  of  water  required  and  1  1.  is  the 
volume  of  the  salt  together  with  the  expansion.  Calculating 
the  volume  of  the  salt  from  its  specific  gravity  the  expansion  is 
obtained.  The  resulting  numbers,  plotted  in  curves  IV  and  V  in 


THE  FLUIDITY  OF  SOLUTIONS 


185 


0     10    20     30    40     50    60    10     80    90    100  110    120    IJO    140    150  160   110    180    190  200  210 

Atomic  Weight 
FIG.  71. — Some  "periodic"  relationships. 


186 


FLUIDITY  AND  PLASTICITY 


Fig.  71,  show  that  in  general  the  salts  which  occupy  the  largest 
volume  in  solution  correspond  to  those  having  the  highest  fluidity 
curve  II,  but  silver  seems  to  be  strongly  exceptional.  Here 
again  we  have  evidence  that  fluidity  is  proportional  to  the  free 
volume.  The  cause  of  the  volume  change  is  also  the  cause  of 
the  negative  curvature. 

Ammonium  iodide  according  to  Getman  (1908)  and  Ranken 
and  Taylor  (1906)  shows  negative  curvature  but  it  goes  into 
solution  with  contraction,  according  to  Schiff  and  Monsacchi. 
There  is  thus  a  lack  of  parallelism  between  the  two  properties  of 
which  one  further  example  may  be  cited.  In  ammonium  nitrate 
solutions,  the  expansion  is  least  in  a  7-weight  per  cent 
solution  and  yet  the  fluidity  is  a  maximum  in  this  solution  at 
some  temperature  between  25  and  40°C.  Since  we  are  dealing 
with  inflected  curves  signifying  simultaneous  dissociation  and 
chemical  combination,  these  anomalies  are  to  be  expected.  The 
limiting  volume  is  continually  changing  and  the  specific  volume  is 
for  that  reason  no  measure  of  the  free  volume.  There  is  need 
for  further  work  in  this  very  important  field. 

Attempts  have  been  made  by  Wagner  and  others  to  assign 
to  each  element  a  specific  viscosity  effect  in  solution.  The  fluidi- 
ties of  nitrates,  chlorides,  and  sulfates  of  certain  metals  in  normal 
solution  at  25°C  are  given  in  Table  XL VII  as  modified  from 
Wagner.  The  table  shows  that  the  fluidity  of  the  nitrates  is 


TABLE  XLVII. — A  COMPARISON  OF  THE  FLUIDITIES  OF  VARIOUS  METALS 
AND  ACID  RADICALS  IN  NORMAL  SOLUTION  AT  25°C  (AFTER  WAGNER) 


NO3 

C13 

S04 

NO3 

Cl 

NO3 

sol 

K  

114.7 

113.3 

101.2 

1.012 

1.133 

K/H  

1.053 

1.081 

0.974 

H  

108.9 

104.8 

102.5 

1.039 

1.062 

K/Na  

1.095 

1.112 

1.112 

Na  

104.7 

101.9 

91.0 

1.027 

1.151 

K/Zn  

1.195 

1.199 

1.239 

Zn  

96.0 

94.5 

81.7 

1.015 

1.175 

K/Mg  

1.201 

1.218 

1.239 

Mg 

95.5 

93.0 

81.7 

1.26 

1.169 

THE  FLUIDITY  OF  SOLUTIONS  187 

always  higher  than  that  of  the  chlorides  and  that  of  the  chlorides 
is  always  higher  than  the  fluidity  of  the  corresponding  sulphate. 
The  ratio  of  nitrate  to  chloride  is  1.02  and  of  nitrate  to  sulphate 
1.14.  We  may  also  compare  the  salts  of  different  metals  joined 
to  the  same  acid  radical  and  thus  get  a  ratio  in  terms  of  one 
metal  taken  for  reference,  as  potassium.  Considering  the  com- 
plex effects  due  to  dissociation,  hydration  and  perhaps  other 
causes,  the  presence  of  even  imperfect  relationships  of  this  kind 
is  remarkable. 


CHAPTER  VI 
FLUIDITY  AND  DIFFUSION 

According  to  Stokes   (1851)  a  sphere  of  radius  r,  impelled 
through  a  fluid  under  a  force  F,  will  attain  the  velocity  v 

•-  £  •  (62> 

This  formula  is  of  fundamental  importance  in  the  study  of  the 
settling  of  suspensions,  diffusion,  Brownian  movement,  the  rate 
of  crystallization  of  solutions,  migration  velocities  and  transfer- 
ence numbers  of  the  ions  and  in  the  conductivities  of  solutions. 
Settling  of  Suspensions. — In  the  case  of  a  falling  sphere,  the 

force  becomes 

4 

F  =  g  7T0r3(p2  -  pi) 

where  pz  and  pi  are  the  densities  of  the  sphere  and  the  medium 
respectively,  so 

v  =  lg(pi  -  Pi>V  (63) 

i7 

This  formula  enables  one  to  calculate  the  speed  of  settling  of 
suspensions.  It  has  been  utilized  in  determining  the  viscosity 
of  very  viscous  liquids,  e.g.,  Tammann  (1898)  and  Ladenburg 
(1907),  for  determining  the  radii  of  the  particles  in  gold  suspen- 
sions, Pauli  (1913),  for  measuring  the  charge  on  the  electron  in 
air,  Millikan  (1910). 

The  Diffusion  Constant.— Sutherland  (1905),  Einstein  (1905) 
and  Smoluchowski  (1906)  have  derived  the  relation  between  the 
diffusion  coefficient  5  and  the  fluidity, 
,  _  RT     9 

~~    N  "fcrr 

where  T  is  the  absolute  temperature,  R  is  the  gas  constant 
(83.2  X  106  c.g.s.  units)  and  N  is  the  number  of  molecules  in  a 
gram  molecule  (70  X  1022).  The  diffusion  coefficient  is  defined 
as  the  quantity  of  solute  diffusing  per  second  through  a  unit 
cube  when  the  difference  in  concentration  between  the  two  ends 
of  the  cube  is  unity.  But  Stokes'  Law  was  derived  for  particles 

188 


FLUIDITY  AND  DIFFUSION 


180 


which  are  spheres  and  having  a  radius  large  in  comparison  with 
the  molecules  of  the  solvent.  If  the  particles  are  so  small  that 
the  free  path  a  of  the  molecules  of  the  suspending  medium  is  ap- 
preciable in  comparison  with  the  radius  of  the  particles,  Suther- 
land (1905),  Cunningham  (1910)  and  Millikan  (1910)  have  shown 
that  Stokes'  formula  becomes 

1  +  — 

'-f-o-e^-  (64) 

where  A  is  a  constant  and  equal  to  about  0.815. 

The  following  table  from  Thovert  (1904)  indicates  that  the 
product  of  the  diffusion  constant  and  the  time  of  efflux  is  approxi- 
mately constant  for  a  considerable  number  of  substances. 

TABLE    XLVTII. — THE    RELATION  BETWEEN   DIFFUSION  AND  VISCOSITY 
(THOVERT) 


Substance 

d  x  io5 

T,  time  of 
efflux 

8XT  X  IO4 

Ether  

3.10 

315 

98 

Carbon  disulfide  

2.44 

405 

99 

Chloroform  

1.50 

660 

99 

Mixture  ethyl  alcohol  and  ether  . 

1.51 

660 

100 

Benzene  

1.24 

790 

98 

Methyl  alcohol  

1.16 

820 

95 

Mixture      ethyl      alcohol      and 

1.03 

950 

98 

benzene  

Water  

0.72 

1,330 

96 

Ethyl  alcohol  

0.59 

1,620 

96 

Turpentine  

0.48 

2,020 

97 

Amyl  alcohol  

0.155 

5,900 

92 

Glycerol  solution  

0.0104 

94,000 

98 

On  the  other  hand,  Oeholm  (1913)  finds  that  8rj  is  not  exactly 
constant  for  a  series  of  alcohols  as  compared  with  water  when 
glycerol  is  the  diffusing  substance.  Oeholm  thinks  that  associa- 
tion and  hydration  will  account  for  the  variations,  at  least  in  part. 

Bell  and  Cameron  have  applied  Poiseuille's  formula  to  diffusion 
through  capillary  spaces  and  find  that  the  distance  y  which  a 
liquid  moves  in  a  given  time  t  is  given  by  the  formula  yn  =  kt, 


190  FLUIDITY  AND  PLASTICITY 

where  n  and  k  are  constants,  and  by  derivation  n  =  2.  The 
formula  is  important  in  dealing  with  diffusion  through  porous 
materials  such  as  soils.  But  in  this  type  of  diffusion,  it  has 
often  been  noticed  that  there  is  a  separation  of  the  components 
of  the  diffusing  substances.  This  subject  will  come  up  for  con- 
sideration later. 

Brownian  Movement. — Einstein  (1906)  has  shown  that  the 
mean  square  of  the  projections  I  of  the  displacement  of  the 
particle  in  time  t  on  the  axis  of  displacement  is 

I2  =  28t 

Substituting  into  this  equation  the  value  of  the  diffusion,  given 
above 

v-RT.*L  f65) 

"    N     3rr 

This  is  the  equation  used  by  Perrin  in  his  brilliant  investigation 
of  the  Brownian  movement. 

The  Velocity  of  Crystallization. — As  a  crystal  forms  in  a 
solution,  the  molecules  of  the  solute  are  drawn  to  the  growing 
face  of  the  crystal.  The  solution  bathing  the  face  of  the  crystal 
has  therefore  a  lower  concentration  of  solute  than  the  main 
body  of  liquid  and  a  process  of  diffusion  must  be  set  up  to  restore 
the  equilibrium.  The  rate  of  crystallization  must  therefore 
depend  upon  the  fluidity  of  the  solution.  Even  in  an  under- 
cooled  liquid,  where  there  is  no  opportunity  for  a  change  in  con- 
centration, the  viscosity  of  the  liquid  retards  the  proper  orienta- 
tion of  the  molecules,  and  crystallization  does  not  take  place 
instantaneously.  H.  A.  Wilson  (1900)  has  demonstrated  that 
the  velocity  is  directly  proportional  to  the  fluidity  of  the  liquid 
at  the  face  of  the  crystal,  according  to  the  formula, 

V  =  a(t  -  t0}<p  (66) 

where  v  is  the  velocity  of  solidification  in  millimeters  per  second, 
a  is  a  constant  and  t0  is  the  temperature  at  which  the  velocity  of 
solidification  is  zero,  i.e.,  the  solidifying  point,  found  by  extrapo- 
lation. This  point  differs  somewhat  from  the  melting-point 
of  the  substance,  being  37°C  for  salol  instead  of  42°C.  This 
signifies  that  the  above  relation  does  not  hold  when  the  amount  of 
undercooling,  and  hence  the  velocity  of  solidification,  is  very 
small.  Since,  Tammann  has  proved  that  purifying  a  substance 


FLUIDITY  AND  DIFFUSION 


191 


always  diminishes  this  region  of  small  velocity,   Wilson  very 
properly  attributes  this  effect  to  impurity. 

Wilson  experimented  with  salol,  benzoic  anhydride,  benzo- 
phenone,  and  azobenzene  confined  in  long  glass  tubes  of  varying 
diameter.  A  thermocouple  was  used  to  get  the  temperature  of 
the  solidifying  surface,  which  was  of  course  different  in  tubes 
of  various  diameters.  How  well  the  observed  and  calculated 
velocities  of  solidification  agree  can  be  seen  in  the  following 
table  for  salol. 

TABLE    XLIX. — THE    VELOCITY  OF   CRYSTALLIZATION  OF  UNDERCOOLED 
SALOL  FROM  WILSON,  MELTING-POINT,  42.0,    t0=  37.0,  a  =  0.065,6 


Temperature, 
degrees 

t  -  <0 

?> 

v,  calculated 

v,  observed 

35 

2 

8.77 

1.15 

1.25 

33 

4 

8.19 

2.14 

2.5 

31 

6 

7.31 

2.90 

3.2 

29 

8 

6.49 

3.40 

3.7 

27 

10 

5.84 

3.82 

3.9 

25 

12 

5.16 

4.05 

4.0 

21 

16 

3.90 

4.08 

4.1 

19 

18 

3.51 

4.13 

4.1 

15 

22 

2.77 

4.00 

4.1 

Since  for  all  of  these  liquids  the  fluidity  is  as  a  first  approxi- 
mation a  linear  function  of  the  temperature,  for  salol  <p  = 
(t  -  9.8)  2.9,  Eq.  (66)  may  be  written 

v  =  at"1  -  pt  +  7 

where  0  —  a(9.8  +  t0)  and  7  =  9.8  at.  Thus,  whereas  it  is 
possible  to  express  the  velocity  of  solidification  as  a  function  of 
the  temperature  only,  it  is  much  simpler  to  express  it  as  a  func- 
tion of  the  fluidity  as  was  done  in  Eq.  (66). 

Migration  Velocity,  Conductivity  and  Transference  Numbers. 
That  the  movement  of  the  ions  under  the  action  of  electrical 
attraction  should  be  dependent  upon  the  fluidity  of  the  solution 
seems  a  natural  inference  from  what  precedes,  and  a  large  num- 
ber of  researches  have  been  devoted  to  the  elucidation  of  the 
exact  relationship.  Since  the  measurement  of  electrolytic  dis- 
sociation depends  upon  this  relationship,  there  can  be  no  question 


192  FLUIDITY  AND  PLASTICITY 

about  its  importance.  One  has  only  to  compare  the  migration 
velocities  of  a  series  of  ions  with  the  fluidities  of  their  chlorides 
in  normal  solution,  as  shown  in  curves  I  and  II  of  Fig.  71  after 
Bredig,  (1894),  to  see  that  there  is  a  definite  relationship  between 
the  two.  In  seeking  an  exact  quantitative  relationship  we  are 
met  again  by  the  awkward  fact  that  water  is  associated  and  elec- 
trolytes in  it  perhaps  always  form  hydrates,  so  that  an  apparently 
simple  aqueous  solution  is  not  simple  in  fact.  The  study  of  molten 
salts,  of  liquid  metals  and  alloys,  and  of  non-aqueous  solutions  for 
this  reason  take  on  a  particular  importance,  but  aqueous  solutions 
have  naturally  received  the  greatest  attention.  The  method 
of  investigation  is  usually  to  change  the  fluidity  of  the  liquid 
by  altering  the  temperature,  concentration  or  pressure  and  to 
observe  the  corresponding  change  in  conductivity. 

As  early  as  in  1851  Wiedemann  investigated  the  viscosity  and 
conductivity  of  various  salt  solutions  of  varying  concentration. 
Wiedemann  calculated  the  value  of  the  ratio  m<p/A.,  where  m  is 
the  percentage  of  salt  and  A  the  conductivity.  He  found  that 
the  ratio  varies  within  narrow  limits  for  each  salt,  e.g.,  for  copper 
sulfate  the  value  varies  from  22.8  to  24.2  when  the  concentration 
is  increased  from  31.17  to  187.02. 

Goure"  de  Villemontee  in  his  monograph  on  Resistance  Electri- 
que  et  Fluidite  has  used  the  results  of  Bouty  and  Bender  to  prove 
that  the  ratio  m^/A  varies  with  the  temperature  in  a  manner 
which  is  the  same  for  all  salts.     (Cf.  Table  L.)     We  have  seen 
that  over  a  small  range  of  temperature 
^  =  ?o(l  +  &) 
so  similarly 

A  =  A0(l  +  at) 

where  a  and  18  are  arbitrary  constants  and  <p0  and  A0  are  the 
fluidity  and  conductivity  at  0°. 

Grossman  (1883)  recalculated  the  results  of  Grotrian  (1876) 

and  found  that  the  ratio  r  is  a  constant  independent  of  the  tem- 
perature, and  the  temperature  coefficients  are  the  same  to  within 
1  per  cent.  Bousfield  and  Lowry  (1902)  using  parabolic  formulas, 
cf.  Eq.  (53a),  in  place  of  the  simpler  linear  ones  given  above, 
found  that  the  constants  in  the  two  formulas  were  the  same 
within  experimental  error. 


FLUIDITY  AND  DIFFUSION 


193 


TABLE    L. — THE     TEMPERATURE    COEFFICIENTS    OF    FLUIDITY    (0)    AND 
CONDUCTIVITY  (a) 


Normality 

ft 

a 

$/<* 

Potassium  chloride 


3.0 

0.0294 

0.0230 

1.3 

2.0 

0.0332 

0.0259 

1.3 

1.0 

0.0372 

0.0291 

1.3 

0.5 

0.0404 

0.0302 

1.3 

Sodium    chloride 

3.0 

0.0390 

0.0279 

1.40 

2.0 

0.0394 

0.0290 

1.37 

1.0 

0.0410 

0.0292 

1.40 

TABLE   LI.  —  FLUIDITY    AND    CONDUCTIVITY    OF  FUSED  SALTS  AND  SALT 

MIXTURES,  AFTER  FOUSSEREAU 

Temperature 
degrees 

, 

A 

,M 

Sodium  nitrate 

305 

0.377 

0.459 

0.821 

320 

0.439 

0.526 

0.799 

329 

0.454 

0.555 

0.818 

340 

0.498 

0.599 

0.832 

355 

0.561 

0.662 

0.848 

Potassium  nitrate 

334 

0.545 

0.631 

0.863 

340 

0.572 

0.661 

0.866 

358 

0.660 

0.790 

0.835 

1  g  mol  Sodium  nitrate   +  1  g  mol  Potassium  nitrate 


232 

0.248 

0.463 

0.534 

242 

0.264 

0.502 

0.526 

266 

0.310 

0.616 

0.504 

287 

0.361 

0.724 

0.499 

304 

0.418 

0.791 

0.528 

313 

0.436 

0.840 

0.519 

332 

0.532 

0.971 

0.548 

348 

0.584 

1.123 

0.520 

359 

0.624 

1.176 

0.530 

194 


FLUIDITY  AND  PLASTICITY 


Foussereau  (1885)  has  examined  the  changes  in  fluidity  and 
conductivity  of  pure  water  with  the  temperature  and  proved 
that  the  conductivity  is  directly  proportional  to  the  fluidity. 
He  has  also  examined  several  fused  salts  and  salt  mixtures  and 
obtained  a  similar  result.  We  reproduce  in  Table  LI  his  results 
for  sodium  nitrate,  potassium  nitrate  and  an  equimolecular  mix- 
ture of  the  two  salts.  It  is  to  be  observed  that  not  only  is  the  ratio 
different  for  the  different  salts  but  the  conductivity  is  relatively 
much  higher  for  the  mixture  than  for  either  of  the  individual  salts. 

Vollmer  (1894)  studied  solutions  of  various  salts  in  methyl 
and  ethyl  alcohols  and  found  the  temperature  coefficients  of 

TABLE  LII. — THE  FLUIDITY  AND  CONDUCTIVITY  OF  TETRAETHYLAMMONIUM 

IODIDE  AT  INFINITE  DILUTION  IN  VARIOUS  SOLVENTS  AT  0°  AND  25°C 

(AFTER  WALDEN) 


Solvent 

jf 

A°; 

#»/A» 

#v 

A2f 

*>/Aco 

Acetone  

252.0 

177.0 

.41 

316.0 

225.0 

1.41 

Propionitrile  

185.0 

129.0 

.43 

242.0 

165.0 

1.47 

Methyl  alcohol  

118.0 

90.0 

.31 

172.0 

124.0 

1.39 

Ethyl  mustard  oil  

118.0 

82.0 

.44 

162.0 

106.0 

1.53 

Acetylacetone 

87  0 

57  0 

52 

128  0 

82  0 

1  56 

Ethyl  alcohol  

55.9 

37.0 

1.51 

92.6 

60.0 

1.54 

Benzonitrile  

51.6 

35.5 

1.45 

80.0 

56.5 

1.42 

Nitrobenzene  

32.6 

25.0 

1.30 

55.0 

40.0 

1.37 

fluidity  and  conductivity  very  nearly  identical.  Walden  (1906) 
has  gone  further  and  proved  that  tp/\  „  is  a  constant  even  when  the 
solvent  is  varied  widely.  He  used  tetraethylammonium  iodide 
in  some  forty  different  organic  solvents  and  found  <p/Aa  =  1.43, 
which  is  independent  both  of  the  nature  of  the  solvent  and  of  the 
temperature.  A  portion  of  his  data  will  serve  to  show  the  nature 
of  the  concordance. 

These  researches  all  point  to  the  conclusion  that  Stokes'  Law, 
with  a  possible  correction  as  already  suggested,  holds  for  the  dif- 
fusion of  molecules  and  ions,  so  that  if  a  given  particle  has  the 
same  size  in  different  solvents  and  at  different  temperatures, 
the  velocity  imparted  by  a  constant  force  will  be  proportional 
to  the  fluidity  of  the  medium. 


FLUIDITY  AND  DIFFUSION  195 

In  ordinary  electrolytic  solutions,  the  dissociation  is  incom- 
plete, hence  it  is  necessary  to  introduce  into  our  formula  the 
dissociation  factor  a  in  order  that  we  may  always  be  dealing 
with  an  equivalent  number  of  ions,  thus 

°^  =  f"  =  const.  (67) 

A  v         Aco 

where  <pv  and  A.v  are  the  fluidity  and  conductivity  at  the  volume 
v.  This  formula  is  of  use  in  obtaining  the  percentage  of  dissocia- 
tion by  the  conductivity  method,  as  indicated  by  Sutherland 
(1902)  Bousfield  (1905)  and  Pissarevski  and  Lempke  (1905). 
Working  with  mixtures  of  alcohol  and  water,  Doroshevskii  and 
Rozhdestvenskii  (1909)  found  that  the  ratio  D<p/A.  was  a  constant 
over  a  considerable  range  of  concentrations  of  alcohol,  D  being 
the  dielectric  constant  of  the  mixture.  There  is  of  course  a  rough 
proportionality  between  the  dielectric  constant  and  the  dis- 
sociating power  of  the  solvent. 

Hartley,  Thomas,  and  Applebey  (1908)  have  applied  the 
Eq.  (67)  to  solutions  of  lithium  nitrate  in  mixtures  of  nicotine  and 
water.  These  mixtures  resemble  ethyl  alcohol  and  water  in 
exhibiting  a  pronounced  minimum  in  fluidity.  The  coefficient 
of  ionization,  calculated  by  formula  (67),  shows  a  maximum. 
The  molecular  conductivities  of  solutions  at  infinite  dilution  of 
the  salt  show  a  minimum,  closely  resembling  the  fluidity  curve, 
whereas  the  molecular  conductivities  of  an  eighth  normal  solution 
show  a  point  of  inflection,  due  to  the  small  ionization  in  the  pure 
solvent. 

Heber  Green  (1908)  has  started  considerable  discussion 
by  the  discovery  that  in  water-sucrose  mixtures,  the  conductivity 
varies,  not  directly  as  the  fluidity,  but 

Am  =  K<f>m  (68) 

where  m  =  0.70  for  lithium  chloride  and  potassium  chloride  and 
0.55  for  hydrochloric  acid.  In  the  case  of  lithium  chloride  no 
single  value  for  m  can  be  found  which  will  give  entirely  satis- 
factory results.  As  a  matter  of  fact,  Washburn  (1911)  has  found 
that  for  the  first  six  sucrose  concentrations,  a  value  of  m  of  0.94 
gives  better  concordance  than  Green's  0.70.  Johnston  (1909) 
has  determined  the  values  of  m  for  a  number  of  other  solvents 
using  the  data  of  Dutoit  and  Duperthius  (1908)  for  sodium 


196 


FLUIDITY  AND  PLASTICITY 


iodide  solutions,  and  he  finds  that  in  no  case  does  the  value  of  m 
depart  from  unity  by  more  than  0.2. 

Johnston  has  calculated  the  value  of  m  for  many  cations 
and  anions  using  different  temperatures  from  0  to  156°,  but 
found  that  no  single  value  could  be  assigned  for  the  hydrogen 
and  hydroxyl  ions.  The  following  table  will  show  the  nature 
of  his  results. 

TABLE    LIII. — THE    RELATION    BETWEEN    THE  CONDUCTANCES  AND  THE 

FLUIDITIES  OF  THE  INDIVIDUAL  IONS  AT  DIFFERENT  TEMPERATURES  C 

(AFTER  JOHNSTON) 


Ion 

A°° 

m 

?/A« 

0° 

P/A, 

100° 

¥>/Ao> 
156° 

K  

40.4 

0.887 

1.39 

1  71 

1  81 

NH4  

40.2 

0.891 

1.40 

1  71 

1  80 

Cl 

41   1 

0  88 

1  37 

1  70 

1  81 

NO3 

40  4 

0  807 

1  39 

1  98 

2  19 

Na  

26.0 

0.97 

2.16 

2.27 

2.31 

^Ca  

30.0 

1.008 

1.88 

1.84 

1.84 

C2H3O2  

20.3 

1.008 

2.77 

2.73 

2.73 

KSO4  

41.0 

0.944 

1.36 

1  51 

1  55 

H  

240.0 

0.234 

0.550 

0.741 

OH  

105.0 

0.535 

0.806 

0.971 

The  slightly  hydrated  ions  K,  NH4,  Cl,  and  N03  have  a 
high  conductivity  and  a  small  value  of  m,  corresponding  to  an 
increasing  ratio  of  ^/A^;  the  presumably  highly  hydrated  ions 
Na,  J^Ca,  C2H3O2,  and  1/2SO4  have  a  low  conductivity,  a  high 
value  of  m  and  a  nearly  constant  ratio  of  ¥>/Aco'  Hydrogen 
and  hydroxyl  are  most  like  the  unhydrated  group  of  ions  in  that 
they  have  a  very  high  conductance  and  a  low  but  rapidly 
increasing  value  of  ip/A^. 

The  explanation  of  these  curious  facts  is  not  at  hand,  but 
apparently  we  must  assume  that  the  conductivity  does  vary 
directly  in  proportion  to  the  fluidity  and  seek  to  explain  the 
inconstancy  of  the  <p/km  ratio  in  the  changing  solvation  of  the 
ions.  The  phenomenon  is  as  if  the  unhydrated  ions  increased  in 
volume  with  the  temperature,  whereas  the  hydrated  ions  do  not. 


FLUIDITY  AND  DIFFUSION  197 

In  the  same  way  the  effect  of  the  addition  of  sucrose  to  lithium 
chloride  solutions  would  be  explained  by  an  increase  in  one  or  both 
of  the  ionic  volumes  due  to  uniting  with  the  sucrose  molecules. 
That  such  a  hypothesis  is  not  improbable,  it  is  well  to  add  that 
C.  H.  Gill  has  found  that  sucrose  does  form  crystalline  com- 
pounds with  the  halides  of  sodium  and  ammonium.  He  did  not 
find  that  lithium  chloride  forms  such  a  compound  but  there  may 
be  a  sufficient  tendency  to  unite  in  solution  to  explain  the  effect 
which  seems  to  be  peculiar  to  sucrose.  Glycerol,  although 
highly  viscous  like  sucrose,  gives  values  of  m  which  are  unity, 
according  to  the  determinations  of  Massoulier  (1900)  as  calcu- 
lated by  Green. 

In  conclusion,  it  may  be  added  that  there  is  no  connection 
between  the  conductivity  and  the  fluidity  of  a  colloidal  solution 
of  gelatine,  as  demonstrated  by  Griffiths  (1896)  (see  also 
Liideking  (1889)).  The  reason  for  this  peculiarity  lies  in  the 
heterogeneous  character  of  colloidal  solutions  as  will  be  more 
fully  discussed  later.  Schweidler  (1895)  has  also  shown  that 
there  is  no  relation  between  conductivity  and  fluidity  in  mercury 
and  certain  amalgams. 

The  Transference  Number. — The  transport  number  nA  is 
expressed  by  the  equation 

IU-T-TA 

AA  +  Ax- 

If  the  equivalent  conductances  of  the  different  ions  change  with 
the  fluidity  at  different  rates,  the  transport  number  must  be  also 
a  function  of  the  fluidity.  We  have  the  two  equations 


and 


whence,  according  to  Washburn  (1911) 

i^v  \  "A-m 

nA  —•  n^ A  ( — ) 

where  N^A  is  the  transport  number  of  the  anion  at  infinite  dilu- 
tion and  m  is  the  exponent  of  Eq.  (62)  for  the  salt. 


CHAPTER  VII 
COLLOIDAL  SOLUTIONS 

If  it  is  highly  important  to  discover  the  relation  between 
fluidity  and  conductivity,  it  is  vastly  more  important  to  have  a 
solution  of  the  numerous  problems  in  connection  with  the 
viscosity  of  colloidal  solutions.  Indeed  it  has  been  said  that 
the  viscometer  is  to  colloid  chemistry  what  the  galvanometer 
is  to  the  subject  of  electricity,  and  Graham  referred  to  the  vis- 
cometer as  a  colloidoscope.  Since  1  per  cent  of  colloid  like  agar 
agar  may  give  water  the  properties  of  a  stiff  solid,  the  advantage 
of  employing  this  property  in  recognizing  the  colloid  state  is 
clearly  apparent. 

A  pure  liquid,  at  a  given  temperature  and  pressure,  can 
have  but  a  single  fluidity,  but  in  our  study  of  liquid  mixtures  we 
have  seen  that  a  mixture  of  liquids  may  have  an  indefinite  number 
of  fluidities  dependent  upon  the  method  of  mixing,  in  other  words, 
upon  the  structure  of  the  liquid.  Since  colloidal  solutions  are 
always  heterogeneous,  they  always  possess  structure,  and  there- 
fore we  have  this  variable  always  entering  into  our  consideration, 
whereas  heretofore  we  have  given  it  but  scant  attention.  There 
is,  however,  every  gradation  from  a  pure  liquid,  to  an  incom- 
pletely mixed  solution,  an  emulsion,  suspension  or  typical  gel. 

The  Two  Types  of  Colloid  Structure. — The  structures  which 
may  occur  are  of  two  kinds,  which  must  be  clearly  differentiated 
from  each  other,  because  they  give  rise  to  phenomena  which  are 
in  some  respects  exactly  opposite,  and  this  is  true  in  spite  of  the 
fact  that  the  two  structures  may  in  certain  cases  merge  into  each 
other. 

In  the  one  case  typified  by  gelatine,  the  structure  requires 
time  to  form  and  the  fluidity  at  a  given  moment  depends  upon 
the  previous  history  of  the  solution.  When  moreover  the 
solution  is  agitated  by  shaking  or  stirring  or  when  it  is  heated, 
the  structure  is  damaged  and  the  fluidity  is  affected.  This 
structure  is  similar  in  results  to  that  which  would  be  produced 

198 


COLLOIDAL  SOLUTIONS  199 

if  an  undercooled  solution  crystallized  out  needle-shaped  crystals 
throughout  the  solution  so  that  flow  of  the  resulting  mass  was 
stopped  except  by  breaking  the  crystalline  structure.  Such  a 
structure  is  a  matter  of  slow  growth,  it  may  be  partially  destroyed 
by  purely  mechanical  means,  and  it  arises  from  forces  which  are 
of  a  polar  nature.  In  view  of  this  analogy  we  may  speak  of  this 
type  of  structure  as  polar,  whereas  the  second  type  is  non-polar. 

In  the  second  type  of  colloidal  solution,  typified  by  clay 
suspensions  those  forces  are  absent  which  bring  about  the 
setting  of  the  gel.  We  have  in  the  typical  case  merely  particles 
of  suspended  solid  which  affect  to  some  extent  the  fluidity  of 
the  solution,  but  as  we  shall  see  the  amount  of  lowering  of  the 
fluidity  is  very  much  less  than  when  the  structure  is  polar  in 
character.  If  the  distribution  of  the  particles  is  uniform,  the 
fluidity  of  the  solution  will  be  independent  of  time,  agitation, 
and  previous  treatment. 

Suspensions. — For  the  simplest  conceivable  case  of  a  solid 
suspended  in  a  liquid,  we  can  imagine  lamellae  of  solid  parallel 
to  the  direction  of  shear  as  discussed  on  page  104.  If  the 
alternating  lamellae  are  sufficiently  numerous,  the  flow  will 
take  place  without  separation  of  the  components,  even  though  the 
fluidity  of  the  one  component  is  zero.  The  fluidity  of  the 
suspension  <f>  is 

<j>  =  cupi  (69) 

where  a  is  the  volume  percentage  of  the  medium  whose  fluidity 
is  <pi  that  is,  the  fluidity  of  the  medium  in  the  limiting  case  will 
be  decreased  in  exactly  the  ratio  which  the  volume  of  the  solid 
bears  to  the  total  volume  of  the  suspension. 

If  the  lamellae  have  an  irregular  surface,  this  law  becomes 
invalid.  If,  for  example,  the  lamellae  are  pierced  by  a  number  of 
fine  pores,  the  fluid  will  fill  these  pores,  yet  the  stream  lines  will 
not  pass  through  the  pores  or  be  appreciably  distorted  by  their 
presence.  The  fluidity  is  then 

*  -  (a  -  d)Vl  (70) 

where  d  is  the  fraction  of  the  total  volume  which  is  pore-space. 

The  ordinary  suspension  consists  of  discrete  particles,  and 

for  the  simplest  case  we  may  consider  a  sphere  suspended  in  a 

fluid  of  its  own  specific  gravity.     The  shearing  of  the  fluid, 


200  FLUIDITY  AND  PLASTICITY 

which  causes  any  cubical  figure  of  the  fluid  to  assume  the  form 
of  a  rhombohedron,  will  cause  the  sphere  to  rotate,  thereby 
assisting  the  flow.  The  stream  lines  are  curved  on  account  of  the 
presence  of  the  sphere,  but  the  sphere  itself  moves  in  a  linear 
direction  and  with  the  velocity  of  the  stratum  of  fluid  which 
would,  if  continuous,  pass  through  the  center  of  the  sphere. 
Spheres  in  the  same  stratum  do  not  approach  each  other  since 
they  all  have  the  same  velocity. 

Spheres  in  different  strata  move  with  unequal  velocity, 
hence  collisions  must  take  place,  depending  upon  the  radii  of 
the  spheres,  their  number  per  unit  volume,  and  also  upon  any 
attraction  or  repulsion  which  may  exist  between  them.  The 


FIG.  72. — Two  spheres  before,  during,  and  after  collision.  The  initial  rota- 
tion of  the  individual  spheres  is  lost  on  collision  and  this  results  in  the  dissipation 
of  energy  as  heat.  In  the  place  of  this  individual  rotation  there  develops  a 
rotation  of  the  system.  It  should  be  noted  that  this  latter  rotation  causes  the 
centers  of  the  spheres  to  move  in  a  transverse  direction,  indicated  by  the  dis- 
tances from  the  dotted  lines. 

surfaces  of  two  spheres  which  are  approaching  each  other  must  be 
moving  in  opposite  directions,  which  are  at  right  angles  to  the  line 
joining  their  centers,  Fig.  72.  The  viscous  resistance  to  this 
shearing  action  which  is  set  up  as  they  approach  will  rapidly 
dissipate  as  heat  their  energy  of  rotation.  In  other  words,  their 
energy  of  rotation  is  converted  into  heat  by  the  "collision"  of 
the  particles. 

The  contact  of  two  particles,  which  are  large  in  comparison 
with  molecular  dimensions,  brings  the  laws  of  ordinary  friction 
into  play.  The  spheres  cannot  rotate  unless  the  torque  exceeds  a 
certain  definite  value,  which  will  become  very  important  when 
we  come  to  consider  plastic  flow.  This  value  depends  upon 
the  pressure  existing  at  their  point  of  contact  normal  to  the 
surfaces  and  this  pressure  in  turn  depends  not  only  on  the  rate 
of  shear  but  on  the  attraction  or  repulsion  which  may  exist 


COLLOIDAL  SOLUTIONS  201 

between  the  particles.  So  when  two  spheres  come  into  contact, 
Fig.  726,  they  must  remain  in  contact  for  a  definite  period 
unless  the  spheres  are  small  enough  to  exhibit  Brownian  move- 
ment. If  the  spheres  were  without  attraction  or  repulsion  for 
each  other,  they  would  become  separated  as  soon  as  their  centers 
have  come  to  be  in  the  same  vertical  plane. 

The  spheres  cannot  rotate  as  individuals  during  the  period 
of  contact  until  the  torque  exceeds  a  certain  minimum  value. 
The  result  is  that  during  the  time  of  contact  the  group  of  spheres 
begin  to  rotate  as  a  whole,  and  they  pass  out  of  the  strata  to  which 
they  formerly  belonged,  Fig.  72c,  and  into  layers  of  different 
velocities.  During  this  period  of  acceleration,  the  liquid  will 
flow  around  the  spheres  and  through  interstices  between  them. 
Thus  other  spheres  tend  to  collide  with  those  already  in  contact 
with  each  other,  after  which  the  combined  mass  tends  to  rotate  as 
a  whole.  When  equilibrium  is  reached  these  clots  will  have  a 
certain  average  size,  depending  upon  the  number,  size,  and  spe- 
cific attraction  of  the  particles. 

For  the  present  purpose,  the  important  thing  to  observe 
is  that  in  the  collisions  of  the  particles  we  have  a  new  source 
of  loss  of  energy,  and  if  these  clots  increase  in  size  and  number 
there  must  come  a  point  when  the  clots  come  in  contact  across 
the  entire  width  of  the  passage.  At  this  point  viscous  flow 
of  the  material  as  a  whole  stops  and  plastic  flow  begins. 

For  a  given  substance  and  volume  concentration,  the  number 
of  collisions  will  be  proportional  to  the  number  of  particles,  which 
varies  inversely  as  the  cube  root  of  the  radius.  But  if  the 
angular  velocity  is  independent  of  the  radius,  the  energy  of 
rotation  will  be  proportional  to  the  square  of  the  radius,  hence 
the  loss  of  energy,  due  to  collisions  will  be  inversely  proportional 
to  the  radius.  This  conclusion,  if  correct,  is  very  important  in 
indicating  that  very  finely  divided  particles  give  comparatively 
viscous  liquids  or  at  higher  concentrations  plastic  solids. 

Bingham  and  Durham  (1911)  have  studied  suspensions  of 
infusorial  earth,  china  clay  and  graphite  suspended  in  water, 
as  well  as  infusorial  earth  suspended  in  alcohol  as  already  referred 
to  on  page  54.  For  each  temperature,  the  fluidity  falls  off  rapidly 
and  linearly  with  the  concentration  of  solid,  so  that  at  no  very  high 
concentration  by  volume  the  fluidity  of  zero  would  be  reached,  as 


202 


FLUIDITY  AND  PLASTICITY 


shown  in  Fig.  73,  for  English  china  clay  and  water.  This  concentra- 
tion of  zero  fluidity  is  independent  of  the  temperature  and  is  the 
concentration  which  serves  to  demarcate  viscous  from  plastic  flow. 


1      I      3      4      5     6      7 
Volume  percentage  Clay 


Fio.  73. — The   fluidity   of   aqueous   suspensions   of   clay   in    water    according 
to  measurements  of  Durham. 


We  are  not  to  conceive  of  a  suspension  of  zero  fluidity  or  infinite 
viscosity  as  .incapable  of  being  deformed,  but  it  would  not  be  per- 
manently deformed  by  a  very  small  shearing  force.  It  remains 
,-m  important  question  which  we  are  unable  to  answer  positively 


COLLOIDAL  SOLUTIONS 


203 


as  yet,  whether  the  viscosity  of  a  suspension  is  independent  of  the 
instrument  in  which  the  measurement  is  made  or  not.  It  seems 
a  necessary  conclusion  that  the  concentration  of  zero  fluidity 
must  be  determined  in  a  long,  narrow  capillary.  The  fluidities  of 
suspensions  follow  the  empirical  formula 


in  which  6  is  the  volume  concentration  of  the  solid  and  c  is  the 
particular  value  of  6  at  which  the  fluidity  of  the  suspension 
becomes  zero.  The  value  of  c  can  vary  only  from  0  to  1,  the 
value  increasing  with  the  size  of  the  particles.  This  equation 
closely  resembles  Eq.  (70)  and  becomes  identical  with  Eq.  (69) 
when  c  =  1. 

In  Table  LI  V  the  fluidities  of  graphite  suspensions  are  compared 

TABLE  LIV.  —  THE  FLUIDITIES  OF  SUSPENSIONS  OF  GRAPHITE  IN  WATER 
AT  DIFFERENT  TEMPERATURES,   (AFTER  BINGHAM  AND  DURHAM) 

C    =5.4   PER   CENT 


Temperature, 
degrees 

Volume 
percentage, 
graphite 

Fluidity 
observed 

Fluidity 
calculated 

Volume 
percentage, 
graphite 

Fluidity 
observed 

Fluidity 
calculated 

30 

0.396 

116.8 

115.7 

1.048 

100.9 

100.7 

35 

0.395 

129.8 

128.3 

1.046 

113.4 

111.7 

45 

0.394 

156.3 

154.8 

1.042 

135.0 

134.8 

55 

0.392 

184.9 

183.0 

1.037 

161.7 

159.5 

65 

0.390 

215.5 

213.1     <l       1.032 

192.1 

185.7 

with  the  values  calculated  by  formula  (71).  The  two  agree 
extremely  well,  which  may  be  due  to  the  fact  that  the  graphite 
suspensions  (aquadag)  are  very  stable,  obviating  trouble  due  to 
settling  out  and  clogging  the  capillary.  That  the  subdivision 
of  the  graphite  is  carried  very  far  is  indicated  by  the  very  low 
value  of  the  concentration  of  zero  fluidity,  c  =  5.4  volume  per 
cent. 

Some  of  the  suspensions  of  sulfur  by  Oden  (1912)  are  plotted 
in  Fig.  74  using  volume  percentages,  taking  1.90  as  the  specific 
gravity  of  sulfur.  These  values  indicate  a  zero  of  fluidity  at 
about  a  25  volume  per  cent  suspension.  Some  of  the  values  are 
not  on  the  curves,  particularly  at  the  high  concentrations;  but  the 


204 


FLUIDITY  AND  PLASTICITY 


measurement  of  the  fluidity  of  suspensions  is  rendered  difficult 
by  the  fact  that  partial  clogging  of  the  capillary  gives  too  low 
fluidities,  and  settling  out  of  the  solid  gives  too  high  fluidities. 
In  reference  to  the  discordant  observation  at  5°C,  Oden  remarks 
that  the  suspension  was  strongly  flocculated. 


50 


40 


20 


\ 


V 


NA 


0 


20 


25 


5  10  15 

Per  Cen-r 

FIG.     74. — Fluidities  of  suspensions  of  sulfur  in  water  at  various  volume  per- 
centages, at  5°,  20°,  and  30°C.      (After  Oden.) 

It  is  interesting  to  find  that  Trinidad  Lake  asphalt,  treated 
with  benzene  gives  suspensions  which  according  to  measurements 
of  Clifford  Richardson  (1916)  indicate  a  zero  fluidity  at  24.6 
volume  per  cent.  The  fluidities  of  the  suspensions  agree  well 
with  our  formula,  which  is  surprising,  since  each  solution  was 
centrifuged  to  remove  that  portion  of  the  suspended  matter 
which  would  not  remain  in  suspension  at  that  particular  con- 
centration. 


COLLOIDAL  SOLUTIONS 


205 


The  curves  of  infusorial  earth  in  water,  page  55,  are  convex  up- 
ward at  the  lower  temperatures  and  convex  downward  at  the 
higher  temperatures.  The  explanation  of  this  behavior  is  not 
known.  Plotting  the  fluidities  and  concentrations  of  "night 
blue"  studied  by  Biltz  and  Vegesack  (1910)  we  find  that  all  of 
those  curves  are  convex  upward,  the  zero  of  fluidity  being  at 


TABLE  LV. — FLUIDITIES  OF  SUSPENSIONS   OF    TRINIDAD   LAKE    ASPHALT 
IN  BENZENE  AT  ABOUT  20°  (AFTER  C.   RICHARDSON) 


Per  cent  asphalt 
by  weight 

Per  cent  colloid 
in  asphalt 

Fluidity  observed 

Fluidity  calcu- 
lated  c  =  34.5 
weight  per  cent 

0 

I 
153.0 

1 

2.54 

153.0 

149 

2 

2.01 

146.0 

144 

5 

2.09 

132.0 

131 

10 

2.73 

104.0 

109 

20 

3.13 

61.0 

64 

30 

4.19 

24.0 

20 

40 

6,51 

11.0 

13 

50 

10.69 

3.1 

about  9.2  weight  per  cent.  Allowing  the  suspensions  to  stand 
for  several  days  causes  a  marked  decrease  in  the  fluidity  as  does 
also  the  purification  of  the  material. 

Woudstra  (1908)  investigated  colloidal  silver  solutions.  In  a 
solution  containing  only  0.0046  per  cent  silver  by  volume, 
the  fluidity  at  26°  was  lowered  4.3  per  cent  so  that  it  seems 
possible  that  a  solution  containing  less  than  1  per  cent  of  silver 
would  have  zero  fluidity!  The  data  are  too  scanty  to  permit 
an  exact  estimation  of  the  zero  fluidity  concentration  and  the 
fluidity-volume  concentration  curve  is  highly  convex  upward. 
With  the  elapse  of  time  and  under  the  influence  of  electrolytes 
colloidal  silver  solutions  coagulate  and  there  is  a  simultaneous 
increase  in  the  fluidity.  This  is  in  accordance  with  our  other 
knowledge  of  the  effect  of  size  of  particle  but  it  is  in  marked 
contrast  to  the  effect  of  "setting"  on  the  fluidity  of  the  polar 
type  of  colloids. 


206  FLUIDITY  AND  PLASTICITY 

Einstein1  and  Hatschek2  have  both  considered  theoretically 
the  case  of  suspensions  of  spherical  particles  at  low  concentrations. 
They  both  arrive  at  the  formula 

H  =  77!  (1  +  kb) 
or 

*  =  rfis  ™ 

where  b  is  the  fraction  of  solid  present  by  volume  and  k  is  a 
constant  for  which  Hatschek  deduced  the  value  of  4.5  and  Ein- 
stein of  1.  The  formula  is  hyperbolic  in  form  while  the  formula 
obtained  from  available  experimental  material  is  linear.  Their 
curve  is  concave  upward,  and  if  it  held  for  high  concentrations 
the  pure  solid  would  have  a  fluidity  of  18  per  cent  (Hatschek) 
to  50  per  cent  (Einstein)  of  the  fluidity  of  the  continuous 
medium,  which  is  absurd. 

Hatschek  states,  "It  is  obvious  that  the  liquid  at  the  upper 
pole  of  each  spherical  particle  moves  with  a  somewhat  greater 
velocity  than  at  the  lower  pole,  which  is  equivalent  to  a  transla- 
tory  movement  of  the  particles  with  a  velocity  equal  to  half  the 
difference  of  the  two  velocities  prevailing  at  the  two  poles." 
He  thus  neglects  entirely  the  rotation  of  the  spheres  and 
assumes  that  they  are  moving  faster  than  the  stratum  of  fluid 
which  would  pass  through  their  centers.  That  these  two  motions 
are  equivalent  is  at  least  not  self-evident.  His  formula  is  ob- 
tained by  the  employment  of  Stokes'  formula  for  a  sphere  moving 
through  a  viscous  medium  without  rotation. 

The  view  is  commonly  held  that  dilute  suspensions  have  a 
viscosity  which  is  very  little  different  from  that  of  the  dispersion 
medium,  but  that  as  the  concentration  is  increased  the  viscosity 
suddenly  increases.  Thus  Ostwald  in  his  Kolloid  Chemie  states, 
"The  curves  and  tables  show  that  at  certain  concentrations  there 
is  a  very  sudden  increase  in  viscosity.  For  silver  and  glycogen 
hydrosols  these  concentrations  are  respectively  about  3.5  and 
30  per  cent."  If  the  fluidity  is  in  fact  linear  as  we  have  indicated 
is  the  case,  the  viscosity  curve  is  hyperbolic.  There  will  naturally 
be  a  rather  sudden  increase  in  viscosity  but  it  has  no  significance. 
The  question  arises,  "Does  the  glycogen  fluidity-concentration 

1  Ann.  der  Physik.,  19,  289  (1906)^ 

2  Kolloid-Zeitschr.,  7,  301  (1901);  8,  34  (1911);  Trans.  Faraday  Soc.  (1913). 


COLLOIDAL  SOLUTIONS 


207 


curve  show  a  sudden  drop  in  fluidity  at  about  30  per  cent?"  The 
glycogen  suspensions  were  studied  by  Botazzi  and  d'Errico  (1906) 
using  two  different  viscometers,  one  from  0  to  20  per  cent  and 
the  second  from  20  per  cent  on.  On  plotting  the  fluidities  we 
find  that  the  values  for  each  viscometer  lie  on  a  straight  line,  but 
the  two  lines  do  not  coincide.  For  the  first  viscometer,  the  fluid- 
ity of  water  is  144.0  and  the  weight  concentration  of  zero  fluidity 
is  27.5,  while  for  the  second  viscometer  it  is  necessary  to  assume 
a  fluidity  for  water  of  77.6  and  a  zero  fluidity  at  41  per  cent 
concentration.  Using  formula  (64)  the  calculated  values  agree 
well  with  the  observed  except  at  45  weight  per  cent  which  is 
beyond  the  concentration  corresponding  to  zero  fluidity,  as  shown 
in  Table  LVI.  Bottazzi  and  d'Errico  give  their  viscosities  as 
times  of  flow,  which  of  course  are  not  proportional  to  the  vis- 
cosities, as  is  so  often  assumed,  so  this  may  perhaps  explain  the 
discrepancy  between  the  two  viscometers.  But  more  work 
needs  to  be  done  on  this  subject  to  definitely  establish  whether 
the  viscosity  of  a  suspension  is  independent  of  the  dimensions 
of  the  instrument  or  not.  At  any  rate  there  is  no  evidence  that 
the  fluidity  of  concentrated  supensions  is  abnormally  low.  In 
fact  these  experiments  lead  to  the  opposite  conclusion. 

TABLE  LVI. — THE  FLUIDITIES  OF  GLYCOGEN  SUSPENSIONS  AT  37°C  (AFTER 
BOTTAZZI  AND  D'ERRICO) 


Per  cent  glycogen 

Fluidity  calcu- 

by  weight 

Fluidity  observed  lated  by  formula 

6 

(71) 

0 

144.0 

144 

1 

138.0 

139 

Viscometer  No.  1 

5 

114.0 

118 

Vl  =  144.0 

10 

86.0 

92 

c  =    27.5 

15 

69.0 

66 

20 

40.0 

40 

20 

40.0 

40 

25 

32.0 

34 

Viscometer  No'.  2 

30 

20.0 

21 

?i  =  77.6 

35 

12.0 

11 

c  =  41.0 

40 

5.0 

2 

45 

2.3 

208 


FLUIDITY  AND  PLASTICITY 


Botazzi  and  d'Errico  obtained  the  viscosity  of  glycogen 
solutions  both  on  raising  the  temperature  and  on  lowering  the 
temperature  to  the  point  of  measurement.  The  difference  was 
hardly  more  than  the  experimental  error,  which  shows  that  the 
fluidity  of  a  suspension  is  not  dependent  on  its  past  history. 
This  is  in  marked  contrast  to  the  behavior  of  polar  colloids.  On 
the  other  hand,  non-polar  colloids  are  very  susceptible  to  the 
effect  of  electrolytes,  even  the  merest  traces  often  causing  a 
change  in  fluidity.  As  a  matter  of  fact  many  suspensoids  show 
a  slow  increase  in  fluidity  on  standing,  due  to  the  gradual  increase 
in  the  size  of  the  particles  on  precipitation,  as  shown  by  Woud- 
stra's  experiments  with  colloidal  silver  suspensions. 

Generally  speaking,  dilute  acids  and  salts  with  an  acid  reaction 
coagulate  suspensions  and  lower  the  fluidity,  whereas  dilute  bases 
and  salts  with  a  basic  reaction  have  a  deflocculating  action. 
Neutral  salts  may  act  in  either  way  or  be  without  effect.  This  is 
shown  in  the  following  table. 

TABLE    LVII. — THE    EFFECT    OF    ELECTROLYTES  ON  THE  FLUIDITY  OF 
SUSPENSIONS  (AFTER  BINGHAM  AND  DURHAM,  (1911)) 


Dispersoid 

Concentra- 
tion disper- 
soid 

Fluidity  of 
suspension 

Substance 
added 

Concentra- 
tion of  elec- 
trolyte 

Fluidity 
with    elec- 
trolyte 

Infusorial  earth  

6.46 

62.1 

KC1 

:80,000 

53.2 

Infusorial  earth  

6.46 

53.2 

NaOH 

:20,000 

58.3 

Graphite  

0.396 

116.8 

KCl 

:  20,  000 

116.9 

Graphite  

0.396 

116.8 

HC.HaO* 

:  20,  000 

64.5 

China  clay  

2.63 

41.5 

KCl 

:40,000 

65.8 

The  decrease  in  the  fluidity  due  to  acids  is  attributed  to  the 
increase  in  cohesion  between  the  particles,  which  results  in  coagu- 
lation. It  is  a  matter  of  common  experience  that  acids  cause 
the  particles  to  cohere  together  and  it  has  already  been  explained 
on  page  200  how  increased  cohesion  decreases  the  fluidity. 

We  need  not  here  discuss  the  reason  why  the  cohesion  of  the 
particles  is  so  much  greater  in  acid  solutions,  although  the 
subject  is  one  of  great  interest  in  the  theory  of  emulsification 
with  its  important  application  in  the  detergent  action  of  soaps. 

Crystalline  Liquids. — Reinitzer  in  1888  first  discovered  that 


COLLOIDAL  SOLUTIONS 


209 


cholesterolbenzoate  melts  at  145.5°  to  an  opalescent  liquid  which 
at  178°  became  suddenly  clear  and  isotropic.  The  optical 
properties  of  this  and  other  substances  of  similar  behavior  was 
carefully  studied  byLehmann.  Schenck  (1898),  Eichwald  (1905), 
Buhner  (1906),  Bose  (1907)  and  Dickenscheid  (1908)  have  studied 


Y 

\ 

0.86 
0.87 
0.&6 

o 

I 

1 

\ 

0 

/ 

1 

1 

/ 

i 

°f 

s 

f 

'  i 

/ 

y 

/ 

i° 

' 

115      120       1Z5 


150      155 


130        135       140       145 

Temperature 

FIG.  75. — Fluidity-temperature  curve  (continuous)  and  specific  volume- 
temperature  curve  (dashed)  of  p-azoxyanisole.  (After  Eichwald  (1905)  and 
Buhner  (1906).) 

the  viscosities  of  these  substances  and  shown  that  these  so-called 
"crystalline  liquids"  have  a  higher  fluidity  than  isotropic  liquids. 
The  specific  volume  of  crystalline  liquids  is  smaller  than  that  of 
isotropic  liquids  of  corresponding  temperature.  In  other  words, 
when  an  anisotropic  liquid  is  heated  to  the  clarifying  point,  there 


210  FLUIDITY  AND  PLASTICITY  • 

is  a  sudden  increase  in  volume  and  decrease  in  fluidity  as  shown 
in  Fig.  75  for  p-azoxyanisole  from  the  measurements  of  Eichwald 
and  Buhner.  As  the  temperature  is  raised,  the  fluidity  increases 
in  a  nearly  linear  manner,  passes  through  a  sharp  maximum, 
and  suddenly  falls  to  the  clarifying  point,  where  there  is  a  dis- 
continuity in  the  curve.  As  the  temperature  is  raised  still 
further,  the  fluidity  again  increases  in  a  linear  manner. 

This  behavior  resembles  that  of  molten  sulfur  which  increases 
in  fluidity  up  to  150°,  where  the  fluidity  is  11.4  according  to  the 
measurements  of  Rotinjanz  (1908).  It  then  suddenly  falls  off 
to  0.0018  at  180°  after  which  the  fluidity  gradually  increases  up 
to  1.14  at  440°. 

Drawing  a  parallelism  between  anistropic  liquids  and  molten 
sulfur,  in  no  way  explains  the  phenomenon,  for  the  behavior  of 
sulfur  is  unexplained.  Bose  regards  anistropic  liquids  merely  as 
emulsions  of  very  long  life.  But  an  emulsion  has  invariably  a 
lower  fluidity  than  a  homogeneous  solution  at  the  same  tempera- 
ture, and  according  to  the  theory  this  must  always  be  the  case, 
so  that  the  emulsion  theory  seems, to  be  excluded.  The  phenome- 
non cannot  be  accounted  for  on  the  basis  of  the  observed  vol- 
ume change,  because  the  volume  of  the  isotropic  liquid  is  greater, 
which  would  lead  to  an  increase  in  the  fluidity.  We  apparently 
have  but  one  explanation  left,  viz.,  that  as  the  anistropic  liquid  is 
heated  to  the  clarifying  point  a  new  molecular  arrangement  is 
formed  which  has  a  much  larger  limiting  volume,  so  that  although 
the  molecular  volume  is  increased  the  free  volume  is  lessened. 
The  same  explanation  would  apply  to  sulfur. 

Emulsions  and  Emulsion  Colloids. — In  our  discussion  of  the 
critical  solution  temperature,  it  was  made  clear  that  the  separa- 
tion of  the  components  of  a  mixture  in  the  form  of  an  emulsion 
is  attended  by  an  increase  in  the  viscosity.  It  seems  probable 
that  this  increase  is  due  to  the  viscosities  in  emulsions  being 
additive,  for  it  follows  of  necessity  that  when  the  viscosities 
are  additive  the  viscosity  will  be  greater  than  in  a  homogeneous 
mixture  of  the  same  composition.  As  in  the  case  of  suspensions, 
there  is  considerable  evidence  that  decreasing  the  size  of  particle 
of  the  disperse  phase  brings  about  a  corresponding  decrease  in  the 
fluidity.  Martici  (1907)  experimented  with  oil-soap  emulsions 
and  found  that  the  fluidity  becomes  less  as  the  drops  become 


COLLOIDAL  SOLUTIONS  211 

smaller.  Buglia  (1908)  has  found  that  the  fluidity  of  milk  is 
lessened  when  the  milk  is  "homogenized"  by  being  squirted 
against  an  agate  plate,  thereby  increasing  the  number  of  fat 
globules.  The  apparent  decrease  in  fluidity  with  emulsification 
finds  excellent  practical  examples  in  the  manufacture  of  solid 
lubricants  and  of  certain  household  products  such  as  mayonnaise, 
"whipped  cream,"  and  beaten  egg  albumen.  In  engine  grease 
less  than  1  per  cent  of  water  emulsified  by  means  of  a  solution  of 
soap  with  mineral  oil  produces  a  salve-like  grease.  Such  bodies 
have  the  properties  of  solids  and  may  also  be  considered  in 
connection  with  the  plasticity  of  solids. 

The  question  inevitably  arises,  "How  is  it  possible  that  water 
with  a  high  fluidity  can  decrease  the  fluidity  of  a  heavy  oil, 
or  air  decrease  the  fluidity  of  albumen,  so  that  the  resulting 
product,  emulsion  or  foam  as  the  case  may  be,  has  the  rigidity 
of  a  solid?  To  answer  this  question  it  is  necessary  to  return  to 
the  consideration  of  our  simple  case  of  lamellae  of  different  liquids 
at  right  angles  to  the  direction  of  shear.  The  theory  that 
viscosities  of  emulsions  are  additive,  will  account  for  the  fluidity 
being  less  than  the  fluidity  of  the  homogeneous  mixture  but  it  will 
in  no  way  account  for  the  case  we  have  here  where  the  fluidity 
of  the  emulsion  is  less  than  the  fluidity  of  either  component. 

As  the  shear  progresses,  it  is  to  be  noted,  Fig.  34,  that  the 
lamellae  are  greatly  elongated.  But  in  immiscible  liquids 
this  thinning  out  of  the  layers  is  opposed  by  the  surface  tension 
which  tends  to  keep  the  surface  area  a  minimum.  If  therefore 
the  shearing  force  is  less  than  the  maximum  force  arising  from 
the  surface  tension,  continuous  deformation  will  not  result. 
There  will  be  a  certain  amount  of  temporary  deformation  but 
this  too  will  disappear  as  soon  as  the  shearing  force  is  removed. 
In  other  words,  the  substance  shows  not  only  rigidity  but  also 
elasticity;  if  the  shearing  force  is  greater  than  "the  elastic  limit," 
continuous  deformation  will  take  place,  but  since  we  are  dealing 
with  immiscible  liquids,  the  lamellae  will  not  be  thinned  out  indefi- 
nitely, but  torn  into  portions  which  will  gather  into  drops  under  the 
influence  of  surface  tension.  Thus  in  an  emulsion,  shear  tends 
to  make  the  droplets  continually  smaller,  and  consequently  to 
raise  the  viscosity.  This  corresponds  to  the  "cold  working" 
of  metals.  This  effect  is  opposed  by  the  spontaneous  coalescence 


212  FLUIDITY  AND  PLASTICITY 

of  the  particles  on  standing,  analogous  to  the  "annealing"  of 
metals,  so  it  appears  that  an  equilibrium  results  and  the  maxi- 
mum in  viscosity  in  emulsions  may  depend  upon  the  rate  of  shear. 

As  the  lamellae  of  the  simple  case,  which  we  have  taken  for 
consideration,  are  broken  up,  the  viscosities  are  no  longer 
strictly  additive.  The  droplets  become  smaller  and  smaller, 
the  surface  tension  becomes  more  and  more  effective,  the  droplets 
become  true  spheres  with  an  inappreciable  amount  of  flow  within 
the  spheres,  so  that  finally  the  distinction  between  emulsion  and 
suspension  disappears. 

We  pass  finally  to  that  class  of  polar  colloids  typified  by 
gelatine,  soap  and  rubber.  In  some  ways  they  are  in  sharp  con- 
trast with  the  type  which  we  have  just  been  considering,  because 
their  viscosity  increases  tremendously  on  standing  and  decreases 
as  a  result  of  shear,  but  they  are  alike  in  the  more  fundamental 
respect  of  exhibiting  the  properties  of  rigidity  and  elasticity. 

It  is  assumed  that  the  process  of  gelatinization  is  the  result 
of  polar  forces  producing  a  network  of  crystals  or  crystal-like 
material  interlacing  throughout  liquid,  without  necessarily 
taking  up  more  than  a  small  portion  of  the  space.  The  solid 
network  performs  the  function  of  the  lamellae  at  right  angles  to 
the  direction  of  shear  in  our  simple  case.  The  cohesion  of  the 
solid  opposes  the  shear  and  gives  rise  to  the  rigidity  of  the  gel. 
The  ability  of  the  solid  to  be  deformed  without  fracture  deter- 
mines its  elasticity.  This  property  of  elasticity  is  enormously 
developed  in  rubber,  and  we  have  seen  that  it  is  noticeable  in 
foams  and  emulsions.  Barus  (1893)  has  noted  the  considerable 
degree  of  elasticity  in  marine  glue  which  may  be  regarded  as 
a  very  viscous  liquid.  It  also  is  of  importance  in  suspensions, 
as  for  example  in  the  manufacture  of  pencils,  the  "leads"  expand 
considerably,  as  they  are  forced  out  of  the  die  previous  to  baking. 

If  gelatinization  is  analogous  to  crystallization,  we  should 
expect  the  viscosity  to  increase  on  standing  and  that  it  would 
be  hastened  by  "seeding"  the  solution  with  a  more  viscous 
colloids.  We  can  readily  see  that  shearing  the  material  would 
result  in  the  destruction  of  the  polar  structure  of  the  material 
and  consequently  in  a  decrease  in  the  viscosity.  We  refer  the 
reader  to  the  rich  material  furnished  by  Garrett  (1903). 

When  a  hydrogel  is  exposed  to  dry  air,   it  loses  moisture 


COLLOIDAL  SOLUTIONS  213 

and  the  structure  gradually  collapses.  But  showing  the  proper- 
ties of  a  true  solid,  it  remains  under  tension,  and  when  placed 
again  in  water,  it  swells  to  approximately  its  former  size,  but  not 
indefinitely,  as  shown  by  Bancroft. 

Increase  in  concentration  of  the  internal  phase  very  naturally 
increases  the  viscosity  of  the  colloidal  solution.  The  addition 
of  non-electrolytes  generally  affects  the  viscosity  in  the  way  that 
we  would  expect  from  the  change  produced  in  the  fluidity  of 
the  external  phase.  Since  the  colloid  may  unite  with  the  water 
to  form  hydrates  or  with  the  non-electrolyte,  we  should  expect 
exceptions  to  the  quantitative  application  of  this  rule.  Electro- 
lytes have  a  similar  effect  on  the  viscosity  of  emulsion  colloids, 
potassium  nitrate,  ammonium  nitrate,  and  potassium  chloride 
which  increase  the  fluidity  of  water  also  increase  the  fluidity  of 
gelatine  solution  according  to  the  measurements  of  Schroeder 
(1903).  Sodium  sulphate,  ammonium  sulphate,  magnesium  sul- 
phate and  lithium  chloride  depress  the  fluidity.  Acids  and 
alkalies  however  first  lower  the  fluidity  and  then  raise  it.  For  a 
more  adequate  account  of  this  complicated  subject  the  reader  is 
referred  to  the  original  papers,  Schroeder,  Pauli,  etc. 

It  has  often  been  a  cause  for  wonder  that  a  gel  which  has  con- 
siderable rigidity  offers  hardly  more  resistance  to  diffusion  than 
does  pure  water.  We  merely  cite  the  names  of  Graham  (1862), 
Tietzen-Henning  (1888),  Voightlander  (1889),  and  Henry  and 
Calugareanu  (1901),  giving  a  single  observation  from  Voight- 
lander to  the  effect  that  a  1  per  cent  solution  of  sodium  chloride 
in  a  1,  2,  and  3  per  cent  solution  of  agar  gave  a  diffusion  constant 
of  1.04,  1.03,  and  1.03  respectively.  Similarly  Liideking  (1889), 
Whetham  (1896),  Levi  (1900)  Garrett  (1903)  and  Hardy  (1907) 
have  found  that  the  conductivity  of  solutions  remains  constant 
during  gelatinization. 

To  understand  these  peculiarities,  it  is  necessary  to  consider 
the  phenomenon  of  seepage  of  a  fluid  through  a  porous  material. 
Suppose,  for  example,  that  we  consider  a  single  pore;  we  must 
assume  that  since  it  is  a  tube  of  capillary  dimensions,  the  flow 
must  follow  the  law  of  Poiseuille  and  be  proportional  to  the  fourth 
power  of  the  radius  of  the  pore.  The  question  arises,  "  What  will 
be  the  effect  upon  the  volume  of  flow  of  substituting  for  the  single 
pore  a  number  of  smaller  pores  whose  total  pore  opening  is  the 


214  FLUIDITY  AND  PLASTICITY 

same  as  that  of  the  single  pore?"  It  is  easy  to  calculate  from 
Poiseuille's  law  that  for  a  given  area  of  pore  opening  the  volume 
of  flow  will  be  directly  proportional  to  the  square  of  the  radius  of 
the  individual  pores,  which  are  assumed  to  be  alike.  If  the  small 
pores  have  a  diameter  which  is  only  0.0001  that  of  the  large  one, 
the  flow  which  takes  place  through  the  large  pore  in  1  minute 
will  require  about  12  years  through  the  multitude  of  pores  having 
the  same  total  area.  The  underlying  principle  on  which  the 
explanation  is  based  is  the  fact  that  each  layer  in  viscous  flow  is 
carried  along  by  the  layer  immediately  below  it,  the  velocities  of 
the  layers  increasing  in  arithmetical  progression.  The  laws  of 
viscous  flow  are  therefore  capable  of  explaining  why  fluids  do 
not  readily  flow  through  jellies  and  other  finely-divided  materials. 

It  is  well  known  that  compact  clay  is  almost  impervious  to 
both  water  and  oils,  and  therefore  they  are  often  associated,  the 
clay  forming  an  impervious  stratum  through  which  the  oil  or  water 
do  not  penetrate.  The  subject  of  pore  openings  is  therefore 
fundamentally  important  to  the  subject  of  the  circulation  of 
water  through  soils  as  well  as  of  their  retention  of  water.  The 
use  of  compact  clay  in  the  cores  of  dams  finds  an  explanation  on 
this  basis. 

When  it  comes  to  a  single  particle  diffusing  through  a  liquid 
impelled  by  electrical  attraction  or  other  force,  the  above  con- 
siderations no  longer  hold  and  the  walls  of  the  pores  offer  no 
serious  resistance,  the  particle  moving  through  the  medium  as  if 
it  alone  were  present,  without  the  surrounding  network. 


CHAPTER  VIII 
THE  PLASTICITY  OF  SOLIDS 

Only  by  the  behavior  of  materials  under  shearing  stresses  are 
we  enabled  to  distinguish  between  a  fluid  and  a  solid.  If  a 
body  is  continuously  deformed  by  a  very  small  shearing  stress, 
it  is  a  liquid;  whereas  if  the  deformation  stops  increasing  after 
a  time,  the  substance  is  a  solid.  This  distinction  is  theoretically 
sharp  like  the  distinction  between  a  liquid  and  a  gas  at  the  critical 
temperature,  but  just  as  a  liquid  may  be  made  to  pass  into  a  gas 
insensibly,  so  a  solid  may  grade  insensibly  into  a  liquid.  Glass 
and  pitch  are  familiar  examples  of  very  viscous  liquids.  Paint, 
clay  slip,  and  thin  mud  in  a  similar  manner  must  be  classed  as 
soft  solids.  According  to  the  experiments  of  Bingham  and 
Durham  (1911)  the  concentration  in  which  the  fluidity  becomes 
zero  under  a  very  small  shearing  force  serves  to  demarcate  the 
two  states  of  matter. 

This  simple  distinction  is  not  always  sharply  drawn  nor  is  its 
significance  thoroughly  appreciated;  and  for  this  reason  much 
labor  has  been  ill-spent  in  the  attempt  to  measure  the  viscosity 
of  solids,  on  the  assumption  that  solids  are  only  very  viscous 
liquids  and  therefore  that  plasticity  and  the  fluidity  of  solids 
are  synonymous  terms.  The  results  are  unintelligible  because 
the  viscosity  as  so  determined  in  various  instruments  is  widely 
different. 

The  views  of  Clerk  Maxwell  expressed  in  his  "  Theory  of  Heat" 
are  especially  noteworthy  and  are  quoted  at  length: 

"  If  the  form  of  the  body  is  found  to  be  permanently  altered  when  the 
stress  exceeds  a  certain  value,  the  body  is  said  to  be  soft  or  plastic  und 
the  state  of  the  body  when  the  alteration  is  just  going  to  take  place  is 
called  the  limit  of  perfect  elasticity.  If  the  stress,  when  it  is  maintained 
constant,  causes  a  strain  or  displacement  in  the  body  which  increases 
continually  with  the  time,  the  substance  is  said  to  be  viscous. 

"When  this  continuous  alteration  of  form  is  only  produced  by  stresses 
215 


216 


FLUIDITY  AND  PLASTICITY 


exceeding  a  certain  value,  the  substance  is  called  a  solid,  however  soft 
it  may  be.  When  the  very  smallest  stress,  if  continued  long  enough, 
will  cause  a  constantly  increasing  change  of  form,  the  body  must  be 
regarded  as  a  viscous  fluid,  however  hard  it  may  be. 

"Thus  a  tallow  candle  is  much  softer  than  a  stick  of  sealing  wax; 
but  if  the  candle  and  the  stick  of  sealing  wax  are  laid  horizontally 
between  two  supports,  the  sealing  wax  will  in  a  few  weeks  in  summer 
bend  under  its  own  weight,  while  the  candle  remains  straight.  The 
candle  is  therefore  a  soft  (or  plastic)  solid,  and  the  sealing  wax  is  a  very 
viscous  liquid. 

"What  is  required  to  alter  the  form  of  a  soft  solid  is  sufficient  force, 
and  this,  when  applied,  produces  its  effect  at  once.  (This  is,  of  course, 
only  relatively  true,  because  plastic  deformation  is  a  function  of  the 
time,  as  will  appear  later.)  In  the  case  of  a  viscous  fluid,  it  is  time  which 
is  required,  and  if  enough  time  is  given  the  very  smallest  force  will 
produce  a  sensible  effect,  such  as  would  be  produced  by  a  very  large 
force  if  suddenly  applied. 

"Thus  a  block  of  pitch  may  be  so  hard  that  you  can  not  make  a  dent 
in  it  with  your  knuckles;  and  yet  it  will,  in  the  course  of  time,  flatten 
itself  out  by  its  own  weight  and  glide  down  hill  like  a  stream  of  water." 

The  italics  and  parentheses  are  ours.  Butcher  (1876)  has 
expressed  views  quite  similar  to  those  of  Maxwell. 

We  may  now  define  plasticity  as  a  property  of  solids  in  virtue 
of  which  they  hold  their  shape 
permanently  under  the  action  of 
small  shearing  stresses  but  they 
are  readily  deformed,  worked  or 
molded,  under  somewhat  larger 
stresses.  Plasticity  is  thus  a  com- 
plex property,  made  up  of  two  inde- 
pendent factors,  which  we  must 
evaluate  separately. 

Reverting  to  our  fundamental 
conception  of  flow  between  two 
parallel  planes  separated  by  a 
distance  dr  and  subjected  to  a  shearing  force  F,  we  have  found 
that  in  a  viscous  fluid 

do  =  <?Fdr 

so  that  if  we  were  to  plot  the  volume  of  flow  through  a  tube  or 
the  rate  of  shear  against  the  shearing  stress,  we  would  obtain 


Shearing  Stress 

FIG.  76. — Typical  flow-shear 
diagram  for  a  series  of  viscous 
liquids. 


THE  PLASTICITY  OF  SOLIDS 


217 


for  a  series  of  fluids  a  family  of  straight  lines  passing  out  from 
the  origin  as  illustrated  in  Fig.  76. 

In  a  plastic  solid,  a  certain  portion  of  the  shearing  force  is 
used  up  in  overcoming  the  internal  friction  of  the  material.  If 
the  stress  is  just  equal  to  the  friction  or  yield  value,  the  material 
may  be  said  to  be  at  its  elastic  limit.  If  the  stress  is  greater  than 
the  friction  /,  the  excess,  F  —  /,  will  be  used  up  in  producing 
plastic  flow  according  to  the  formula 

dv  =  n  (F  -  /)  dr  (73) 

where  ju  is  a  constant  which  we  will  call  the  coefficient  of  mobility 


Shearing  Stress 

FIG.  77. — Flow-shear  diagram  of  a  plastic  solid. 

in  analogy  to  the  fluidity  of  liquids  and  gases.  If  we  were  to 
plot  the  volume  of  flow  against  the  shearing  stress  we  would 
again  obtain  a  straight  line  for  a  given  material  but  it  would 
not  pass  through  the  origin,  ABC  Fig.  77. 

It  is  easy  now  to  see  why  the  "viscosity"  of  plastic  substances, 
as  measured  in  the  usual  way  for  liquids,  is  not  a  constant. 
Referring  to  the  figure,  if  we  take  two  determinations  of  the 
flow  A  and  B,  we  see  that  they  correspond  to  entirely  different 
viscosity  curves  OD  and  OE. 

When  the  stress  is  not  equal  to  the  yield  value,  the  material 
undergoes  elastic  deformation  and  an  opposing  force  arises 


218  FLUIDITY  AND  PLASTICITY 

which  would  restore  the  body  to  its  original  shape  if  it  were 
perfectly  elastic,  as  soon  as  the  stress  was  removed.  On  the 
application  of  the  stress,  the  restoring  force  is  first  zero,  then 
gradually  increases  to  a  maximum,  when  at  last  the  flow  causes 
the  strain  to  disappear  as  fast  as  it  is  produced. 

The  elasticity  e  of  a  solid  may  be  calculated,  according  to 
Morris-  Airey  (1905),  from  the  fundamental  formula 

ds  =  eFdr  (74) 

where  ds  is  the  distance  which  one  plane  of  the  material  is  sheared 
in  reference  to  another  plane  which  is  separated  from  it  by  a 
distance  dr,  each  being  subject  to  the  shearing  force  F.  Morris- 
Airey  has  applied  this  formula  to  tubes  of  circular  cross-section 
filled  with  gelatine  and  obtained  the  rigidity1  f  which  is  the 
reciprocal  of  the  elasticity 


where  V  is  the  volume  of  the  temporary  deformation.  It  is 
assumed  that  the  solid  is  incompressible.  The  analogy  of  this 
formula  with  that  of  Poisuille  is  striking. 

THE  METHODS  FOR  MEASURING  THE  FRICTION  AND 
MOBILITY 

To  determine  the  two  quantities,  friction  and  mobility,  which 
go  to  make  up  the  plasticity  of  a  material,  i.e.,  to  locate  the  curve 
in  Fig.  77,  it  is  necessary  to  make  at  least  two  measurements  of 
the  flow,  using  different  stresses.  We  may  use  the  tube  method 
(Bingham  (1916)),  the  torsion  method  (Perrott  (1919)),  or  we 
may  observe  the  flow  in  a  rod  under  traction  or  torsion,  the 
flow  of  a  cylinder  under  axial  compression,  the  rate  of  bending 
of  a  horizontal  beam  of  the  material  under  its  own  weight,  or  the 
flow  of  a  freely  descending  stream  of  material,  (Trouton  and 
Andrews  (1904)).  Still  other  methods  have  been  suggested 
such  as  the  rate  of  decay  of  vibrations  in  solid  bodies,  (Kelvin 
(1865)  and  others). 

The  friction  is  most  easily  obtained  by  the  graphical  method, 

PP 

plotting  the  rates  of  flow  V/t,  against  the  shear,  F  —  ~$\    an<^ 

1  The  assumption  which  is  sometimes  made  that  the  rigidity  is  the  re- 
ciprocal of  the  mobility  is  incorrect. 


THE  PLASTICITY  OF  SOLIDS 


219 


extrapolating  the  curve  to  the  axis;  the  value  of  the  intercept 
will  evidently  be  the  friction.  We  may  also  use  the  algebraical 
method.  In  either  case  at  least  two  measurements  of  the  rate 
of  flow  V\/ti  =  vi  and  Vz/h  =  v2  are  necessary  corresponding  to 


o.oos 


I  / 
I  / 
II 

II 

II 


II 


1500 


2000 


0  500  1000 

Shearing  Stress 

FIG.  78. — Flow-shear  curves  of  a  certain  paint,   using  capillaries  of  varying 
length  and  radius. 

the  shears  F\  and  Fz,  respectively.  Assuming  that  the  mobility 
is  independent  of  the  rate  of  flow,  Eq.  (73)  integrated  in  Eq.  (89) 
gives  us 

/  =  —  (76) 


220 


FLUIDITY  AND  PLASTICITY 


The  following  table,  taken  from  the  work  of  Bingham  and 
Green  on  paints,  proves  the  validity  of  the  general  law  of  plastic 
flow  expressed  in  Eq.  (73).  The  friction,  when  expressed  in 
terms  of  shear — and  not  in  terms  of  pressure — is  nearly  constant 
and  not  a  function  of  the  dimensions  of  the  capillary.  It  is  a 
fact,  however,  that  the  rate  of  flow  is  not  directly  proportional  to 
the  shear,  when  the  shear  is  too  small,  but  when  the  shear  is  suffi- 
ciently high  the  relation  becomes  linear,  as  is  proved  by  plotting 


VOLUME  'PERCENTAGE:  CLAY 


FIG.  79. — The  relation  of  fluidity  and  friction  to  volume  concentration  of  solid 
in  clay  suspensions. 

the  values  in  the  table,  Fig.  78.  The  table  also  indicates  that  the 
mobility  is  a  constant  independent  of  the  rate  of  flow  or  of  the 
dimensions  of  the  capillary.  The  reason  for  the  rate  of  flow- 
shear  curve  not  being  linear  as  the  rate  of  flow  is  decreased  will 
be  considered  when  we  come  to  discuss  the  theory  of  plastic 
flow. 

By  measuring  the  fluidity  of  suspensions  containing  increasing 
amounts  of  solid  in  suspension,  Bingham  and  Durham  found 
it  possible  to  obtain  a  concentration  which  would  possess  zero 
fluidity  when  the  shear  was  very  small.  Conversely,  by  measur- 
ing the  friction  of  suspensions  containing  decreasing  amounts  of 
solid,  it  is  possible  to  find  a  concentration  which  would  have 


THE  PLASTICITY  OF  SOLIDS 


221 


zero  friction,  Fig.  79.  Evidently  these  two  concentrations  are 
identical,  and  the  concentration  of  zero  fluidity  or  of  zero  friction 
is  a  fundamental  constant  of  the  material  giving  important 
information  in  regard  to  its  nature,  it  being  intimately  related  to 
the  size  of  the  particles  and  to.  the  adhesion  between  them. 
The  flow  of  a  given  material  is  defined  completely  by  a  knowl- 
•edge  of  the  friction  and  mobility,  but  when  the  concentration  of 
the  suspension  is  changed,  a  knowledge  of  the  concentration  of 
zero  fluidity  is  necessary  in  order  to  estimate  the  effect  produced 


0  25  50  75  10O 

VOLUME  PERCENTAGE;  CLAY 

FIG.  80. — Relation  between  mobility  and  volume  concentration  of  solid  in  clay 
suspensions. 


upon  the  friction  and  mobility.  It  therefore  seems  probable  that 
the  concentration  of  zero  fluidity  is  a  variable  which  is  inde- 
pendent of  both  the  friction  and  the  mobility. 

Finally,  we  may  add  that  the  mobility  of  suspensions  de- 
creases very  rapidly  with  increasing  concentration  of  solid 
as  indicated  by  measurements  of  the  author  which  are  plotted  in 
Fig.  80.  Clay  suspensions  were  used  having  a  concentration  of 
zero  fluidity  of  19  per  cent  by  volume.  The  mobility  starts  at  a 
very  large  but  undetermined  value  and  quickly  falls  to  a  Very 
small  value  in  a  concentration  of  about  50  per  cent  by  volume. 
The  friction  on  the  other  hand,  starts  at  zero  in  the  19  per 
cent  mixture  and  rises  steadily  and  in  an  apparently  linear  man- 
ner as  the  concentration  is  increased  as  seen  in  Fig.  79. 


222 


FLUIDITY  AND  PLASTICITY 


TABLE  LVIII. — FRICTION  AND  MOBILITY  OF  A  PAINT  AS  MEASURED  BY 

BlNGHAM   AND  GREEN1 


Num- 
ber of 
obser- 
vation 

V-t  centi- 
meters per 
second 

Pressure 
grams 
per 
square 
centi- 
meter 

_       PR 

F  --5T 

dynes 
per 
square 
centi- 
meter 

/ 
dynes 
per 
square 
centi- 
meter 

F-f 

H 

Remarks 

Obser- 
vations 
used    in 
calcu- 
lations 

1 

0.0005836 

670.8 

1030.7 

98.2 

938.7 

0.260)      Capillary  S 

1  and    2 

2 

0.0004557 

537.8 

826.3 

84.6 

734.3 

0.260 

r  =  0.014486 

2  and    3 

cm 

3 

0.0003344 

409.3 

628.9 

75.9 

536.9 

0.261 

I  =  4  .  620  cm 

3  and    4 

4 

0.0002133 

277.5 

426.4 

[66.0] 

334.4 

0.267 

4  and    o 

5 

0.0001661 

225.6 

346.6 

[57.7] 

254.6 

0.273 

5  and    6 

6 

0.0001019 

152.9 

234.9 

7 

0.002424 

670.2 

1458.6 

101.0 

1366.6 

0.253 

Capillary  VI 

8 

0.001912 

538.5 

1171.9 

85.4 

1079.9 

0.254 

r  =  0.020805 

9 

0.001418 

409.5 

891.2 

87.5 

799.2 

0.255 

I  =  4  .  684  cm 

9  and  10 

10 

0.0008987 

274.3 

596.9 

[65.6] 

504.9 

0.256 

10  and  11 

11 

0.0004164 

143.3 

311.8 

[53.7] 

219.8 

0.272 

11  and  12 

12 

0.0002880 

106.7 

232.2 

13 

0.004638 

671.7 

1723.0 

81.6 

1631.0 

0.246 

Capillary  III 

13  and  14 

14 

0.003678 

539.1 

1382.9 

93.2 

1290.9 

0.246 

r  =  0.02450 

14  and  15 

15 

0.002726 

409.0 

1049.2 

85.1 

957.2 

0.246 

I  =  4.681  cm 

15  and  16 

16 

0.001758 

275.6 

706.9 

[75.1] 

614.9 

0.247 

16  and  17 

17 

0.0008267 

145.1 

372.2 

[63.5] 

280.2 

0.255 

17  and  18 

18 

0.0005856 

110.0 

282.2 

The  average  friction  used  in  calculating  the  mobility  is  92 . 0  dynes  per  square  centimeter, 
which  gives  an  average  mobility  of  0 . 257.  When  the  rate  of  flow  V/t  is  too  small,  the 
friction  becomes  smaller,  as  seen  in  the  table  and  the  last  two  values  for  each  capillary  may 
well  be  neglected. 

The  Capillary  Tube  Method. — Unless  the  conditions  of 
flow  are  carefully  chosen,  the  friction  constant  does  not  manifest 
itself,  or  at  any  rate  the  amount  of  shear  is  not  a  linear  function 
of  the  shearing  stress.  This  departure  from  linearity  is  very 
often  shown  at  the  low  rates  of  shear  as  indicated  in  Fig.  76 
by  the  curve  FG. 

This  peculiarity  is  not  fully  understood  at  present  and  the 
worker  will  do  well  to  avoid  anxiety  in  regard  to  it  by  choosing 
the  conditions  as  nearly  ideal  as  possible  so  that  the  flow  will 
be  a  linear  function  of  the  shearing  stress. 

Nevertheless  the  cause  of  the  above  peculiarity  must  be 
investigated  in  detail  if  we  are  to  understand  fully  the  nature 

1Proc.  Am.  Soc.  for  Test.  Mats.  (1919). 


THE  PLASTICITY  OF  SOLIDS  223 

of  plastic  flow  and  it  has  already  had  the  attention  of  Bucking- 
ham (1921).  In  plastic  material  confined  between  two  parallel 
planes  of  indefinite  extent  which  are  being  sheared  over  each 
other,  the  shearing  stress  F  will  be  identical  at  every  point.  But 
in  flow  through  a  capillary  tube  according  to  Buckingham 
this  is  not  the  case;  the  shear  increases  continually  from  the 
center  of  the  capillary  outward  and  only  at  a  certain  distance  r0 
does  the  shearing  force  become  sufficient  to  overcome  the  friction. 
Therefore  the  material  at  the  center  of  the  capillary  moves  as  a 
solid  plug  with  the  velocity  v0,  and  beyond  the  radius  r0  the  mate- 
rial moves  in  telescoping  layers.  This  results  in  the  flow  not 
being  a  linear  function  of  the  pressure. 

But  there  are  other  possible  causes  of  the  peculiarity  which 
may  be  mentioned  here.  The  plastic  material  next  to  the 
wall  may  have  a  lower  concentration  of  solid  than  elsewhere 
resulting  in  apparent  slippage.  Or  the  shearing  stress  may  cause 
the  liquid  to  flow  between  the  particles  of  solid,  seepage. 

Buckingham  suggests  that  the  friction  between  the  particles 
during  flow  may  not  be  the  same  as  the  static  friction.  It 
seems  further  possible  that  the  friction  will  need  further  definition 
when  the  individual  particles  of  the  plastic  material  are  of  very 
different  sizes.  We  shall  at  first  assume  that  slippage  and 
seepage  are  both  absent  and  that  the  particles  of  solid  are  uni- 
formly small. 

The  total  force  producing  the  flow  through  a  capillary  tube 
is  wPR2  and  since  there  is  no  acceleration,  this  is  opposed  by  a 
force  in  the  opposite  direction  2-n-RlF.  If  p  is  the  pressure 
which  is  used  up  during  the  flow  in  overcoming  the  friction,  the 
friction  /  is  defined  by  the  equation 


(77) 

Ok 

It  also  follows  that 

(78) 

(79) 

ait 

Since  the  speed  decreases  as  the  radius  increases,  Eq.   (73) 
becomes 


224  FLUIDITY  AND  PLASTICITY 

where  v  is  the  velocity  parallel  to  the  axis  at  the  radius  r.  The 
speed  of  the  material  in  the  variable  region  is  obtained  by 
integrating  Eq.  (80)  from  r  =  R  to  r  =  r  or 


(81) 

The  speed  of  the  solid  plug  is  obtained  by  making  r  =  r0  in 
Eq.  (81),  and  is  after  simplifying 

U*-»&r+7-fM)  (82) 

The  volume  of  flow  per  unit  of  time  is  V/t  and 

v     CR 

-  =        2-nrvdr 
t       Jo 

or  using  Eqs.  (81)  and  (82) 

v  CR 

-  =  *r02i>o  +  27T        rvrdr.  (83) 

t  Jr, 

But  from  Eqs.  (78)  and  (82) 

If*       ,D\ 


and  from  Eq.  (81) 

27r  f    rMr  =  27TM    f   |  ^  (fl2r  -  r3)  -  f(Rr  -  r2)  1  dr 


-  -  -- 

l6^         6         41  \       2       4/  2 

and  introducing  the  value  of  r0  from  Eq.  (78),  we  have 


Introducing  these  values  of  the  separate  terms  of  Eq.  (83)  and 
simplifying,  Eq.  (83)  becomes 

V  /R*P      R3f  . 


_  4    /2(f\   _   J^  /2g\«-| 
t         81    L         3   \R/       3P3  \«/  J 
and  now  introducing  the  value  of  p  given  by  Eq.  (77), 


THE  PLASTICITY  OF  SOLIDS  225 

For  large  values  of  the  applied  pressure,  the  last  term  of  Eq. 
(86)  becomes  very  small  and  the  curve  becomes  very  nearly 
linear  and  coincident  with  its  asymptote 


-=£>-!')•  m 

The  curve  rises  above  the  asymptote  as  the  applied  pressure 
becomes  very  small,  but  it  crosses  the  pressure  (or  shearing  stress) 
axis  when  P  =  p  (or  F  —  /)  .  On  differentiating  Eq.  (86)  in 
respect  to  the  pressure  one  finds  that  the  slope  of  the  curve 
vanishes  when  P  =  p,  hence  the  curve  is  tangent  to  the  axis. 

The  intercept  of  the  asymptote  is  thus  4/3  of  the  true  friction 
which  would  be  obtained  by  other  methods  as,  for  example, 
plastic  material  confined  between  parallel  planes  which  are  being 
sheared  over  each  other.  If  in  practice  conditions  may  be  con- 
trolled so  that  all  of  the  observed  points  lie  on  a  straight  line,  it 
will  mean  that  all  of  the  observed  points  lie  on  a  straight  line,  the 
capillary  in  telescoping  layers,  the  term  p4/3P3  being  negligible. 

Were  we  to  assume  that  the  material  throughout  the  capillary 
flows  in  telescoping  layers  for  all  shearing  stresses  above  /,  we 
will  obtain 


=         -/)  (89) 

which  differs  from  Eq.  (88)  in  having  /  in  place  of  4/3/.  It  is 
highly  desirable  that  some  one  measure  the  friction  both  by  the 
capillary  tube  method  and  other  methods  using  a  given  material, 
to  make  sure  that  they  give  identical  values  for  the  friction. 

.Not  being  able  to  reproduce  satisfactorily  the  data  of  Bingham 
and  Green,  Buckingham  has  attempted  to  allow  for  slippage.  If 
there  is  a  thin  layer  of  viscous  liquid  of  thickness  e  separating  the 
plastic  material  from  the  wall,  it  will  increase  the  velocity  of  the 
plastic  material  by  the  amount  epF,  hence  the  increase  in  the 
volume  of  flow  per  unit  of  time  over  that  given  by  Eq.  (88)  is 
irR2e<pF  approximately.  But  at  present  it  is  not  certain  that  there 
is  slippage  after  the  flow  is  established  by  increasing  the  shearing 


226  FLUIDITY  AND  PLASTICITY 

stress  somewhat  above  the  friction,  so  we  have  no  idea  as  to  how 
the  value  of  e  may  vary  with  the  rate  of  shear,  and  the  equation 
becomes  unmanageable.  Fortunately  by  using  the  higher  rates 
of  shear  we  can  apparently  always  obtain  the  simple  linear 
relationship.  If  later  experiments  prove  that  this  is  not  the  case 
it  will  be  time  to  use  the  more  complex  formulas. 

The  Traction  Method. — Trouton  has  discovered  that  the  rate 
of  flow  in  a  rod  of  material  subjected  to  traction  is  not  propor- 
tional to  the  tractive  force  T,  but  analogously  to  Eq.  (73) 

dv  =  \(T  -  t)dr  (90) 

where  X  is  the  coefficient  of  plastic  traction,  and  t  is  a  tractive 
friction  constant.  The  value  of  i  may  be  found  by  plotting 
the  elongation  dv/dr  against  the  tractive  force  and  extrapolating 
the  curve  to  the  axis.  Trouton  has  obtained  values  of  X  for 
pitch  of  2.3  X  10-10  and  for  shoemakers'  wax  of  1.7  X  10~7.  To 
obtain  the  relation  between  the  coefficient  of  plastic  traction  and 
the  mobility,  we  note  that  the  tractional  force  applied  to  a  rod 
may  be  resolved  into  two  equal  shearing  stresses  at  right  angles 
to  each  other  and  at  45°  to  the  direction  of  traction.  The  value 
of  either  shearing  stress  is  one-third  of  that  of  the  tractive  stress, 
hence  the  friction  is  one-third  of  the  tractive  friction  and  the 
mobility  is  one-third  of  the  plastic  traction  coefficient  as  shown  in 
Table  LX. 

The  Torsion  Method. — Trouton  applied  a  constant  torque 
to  the  ends  of  a  cylinder  or  tube  of  substance  and  observed  the 
relative  motion  of  the  ends.  He  found  that  rods  which  were 
carefully  made  could  be  twisted  almost  indefinitely,  provided 
that  they  were  maintained  in  a  horizontal  position.  The  motion 
was  fastest  when  the  stress  was  first  applied  but  the  angle  of 
twist  per  unit  length  U  soon  became  a  linear  function  of  the  time. 
Conversely  when  the  stress  was  removed,  the  bar  started  to 
twist  in  the  opposite  direction.  He  made  the  experiment  of 
removing  weights  at  such  a  rate  that  the  rod  would  not  move  in 
either  direction,  and  found  that  the  weights  remaining  were 
inversely  proportional  to  the  time  elasped.  This  kind  of  elastic 
recovery  was  found  to  be  present  in  glass  and  sodium  stearate. 
Trouton  does  not  seem  to  regard  his  materials  as  solids  but  he 
makes  it  very  clear  that  the  angular  velocity  is  not  directly  pro- 


THE  PLASTICITY  OF  SOLIDS  227 

portional  to  the  torque,  and  there  is  a  very  considerable  magni- 
tude to  the  value  which  can  be  assigned  to  the  friction  in  his 
experiments  with  pitch. 

Trouton  assumes  from  symmetry  that  any  two  planes  in  the 
material,  lying  at  right  angles  to  the  axis  of  the  cylinder,  move 
over  each  other,  about  the  common  axis,  remaining  plane  all  the 
while. 

Let  8x  be  the  distance  apart  of  the  two  planes,  and  6w  be  the 
relative  angular  velocity  of  the  planes,  then 


where  F  is  the  torque  applied  and  /  is  the  force  used  up  in  over- 
coming the  friction,  obtained  by  extrapolation. 
Thus  for  a  solid  cylinder  we  have 


and  for  a  tube  of  material  this  becomes 


where  Ri  and  Rz  are  the  external  and  internal  radii  respectively. 
Trouton  proved  the  validity  of  the  fourth-power  law  by  using 
two  cylinders  of  pitch  whose  radii  were  0.36  and  0.67  cm  and 
obtained  mobilities  of  1.01  X  10~u  and  0.99  X  10~u  respectively 
which  is  excellent  agreement. 

The  Sagging  Beam  Method.  —  The  rate  of  sagging  U  of  a  rod 
at  its  center  is  found  to  be 

,7         5     gmL*  , 

-384  "XT 

where  m  is  the  mass  of  the  rod  between  the  supports,  L  is  its 
length  and  I  is  the  moment  of  inertia  of  the  cross-section  of  the 
rod,  and  g  is  the  gravitation  constant.  This  does  not  take 
account  of  the  friction. 

In  order  to  prove  that  the  rate  of  sagging  of  a  beam  varies  as 
the  fourth  power  of  the  length,  Trouton  measured  the  times  T 
which  beams  of  different  lengths  required  to  sag  a  certain  dis- 
tance. Table  LIX  shows  that  TL4  is  very  nearly  constant. 


228 


FLUIDITY  AND  PLASTICITY 


TABLE  LIX. — EXPERIMENTS  ON  THE  SAGGING  OF  A  ROD  OF  PITCH  AT  15C 
DEMONSTRATING  THAT  THE  TIME  T  REQUIRED  TO  SAG  A  GIVEN  DIS- 
TANCE VARIES  INVERSELY  AS  THE  FOURTH  POWER  OF  THE  DISTANCE 

L   BETWEEN    THE    SUPPORTS    (AFTER    TROUTON) 


L 

T 

TL* 

33 

14.6 

1.7X107 

30 

18.5 

1.5 

27 

30.4 

1.6 

24 

47.0 

1.6 

That  the  different  methods  agree  with  each  other  is  shown 
in  Table  LX. 

TABLE  LX. — A  COMPARISON  OF  THE  COEFFICIENTS  OF  PLASTIC  TRACTION 
AND  MOBILITY  AS  DETERMINED  BY  VARIOUS  METHODS  (AFTER  TROUTON) 


Substance 

X 

Method 

- 

Method 

n/\ 

Pitch  I 

2  3X10-  J 

Traction 

7   1  X10~u 

Torsion 

3  07 

Pitch  II  

2.8X10-  > 

Traction  

1.0X10-™ 

Torsion... 

3.60 

Pitch  II  

3.  OX  10'  ' 

Sagging  

3.30 

Pitch  and  tar  I  

.8X10-» 

Traction  

2.4X10-10 

Torsion  .  . 

3.07 

Pitch  and  tar  II  

.5X10-  » 

Traction.... 

4.5X10-10 

Torsion... 

3.04 

Shoemaker's  wax  

.9X10- 

Traction  

S.OXIO"7 

Torsion  .  . 

2.95 

Pitch  and  tar  3:1  III  

.3X10- 

Sagging  

3.8X10-* 

Efflux  

3.25 

Pitch  and  tar  3:1  IV  

.1x10- 

Descending 

column  

3.6X10-5 

Efflux.... 

2.91 

THE  THEORY  OF  PLASTIC  FLOW 

A  plastic  solid  is  made  up  of  particles  which  touch  each  other 
at  certain  points.  The  spaces  between  the  particles  may  be 
empty  or  it  may  be  filled  with  gas,  liquid,  or  amorphous  solid. 
Flow  necessitates  the  sliding  of  these  particles  the  one  over  the 
other  according  to  the  ordinary  laws  of  friction,  so  long  as  the 
particles  are  large  enough  so  that  their  Brownian  movement  is 
negligible.  It  is  by  no  means  necessary  that  the  particles  be 
touching  at  the  maximum  number  of  points,  corresponding  to 
"close-packing."  As  a  matter  of  fact,  close-packing  of  the 
particles  prevents  flow  from  taking  place.  It  is  merely  necessary 
that  the  particles  touching  each  other  form  arches  capable  of 
carrying  the  load,  as  already  indicated  on  page  201 .  It  is  evident 


THE  PLASTICITY  OF  SOLIDS  229 

that  as  aggregates  of  particles  are  formed  in  the  process  of 
collisions,  and  the  size  of  these  aggregates  increases  as  the 
concentration  of  solid  increases,  there  must  come  a  time  when 
such  aggregates  or  clots  will  touch  each  other  and  form  an  arch  or 
bridge  across  the  space  through  which  the  flow  is  taking  place. 
At  that  concentration  the  friction  will  have  a  finite  value,  and  the 
material  may  be  said  to  have  a  structure  just  as  was  the  case  of 
the  jelly  or  foam  already  considered. 

The  pore  space  may  vary  between  very  wide  limits,  but 
if  the  suspended  particles  are  assumed  to  be  uniform  spheres,  it 
can  easily  be  calculated  that  cubical  close-packing,  would  leave 
a  pore  space  of  1  —  Tr/6  or  47.64  per  cent  by  volume,  irrespective 
of  the  size  of  the  particles.  It  is  possible  to  get  the  particles 
still  closer  together  until  with  tetrahedral  close-packing,  which 
we  have  in  a  pile  of  cannon-balls,  the  pore  space  is  1  —  7r/3\/2  or 
25.96  per  cent  by  volume,  but  in  this  case  the  particles  are 
interlocked  and  no  true  flow  is  possible  but  rupture,  with  dis- 
integration of  the  particles.  When  the  pore  space  is  roughly  50 
per  cent,  the  mobility  is  zero,  and  it  is  only  as  the  pore  space 
is  in  excess  of  this  figure  that  the  mobility  has  a  finite  value. 
This  excess  pore  space  thus  plays  a  role  which  is  analogous  to 
the  free  volume  of  liquids. 

As  there  is  a  minimum  in  the  allowable  pore  space  in  a  plastic 
solid,  so  there  is  a  maximum,  for  as  the  pore  space  increases  the 
substance  finally  ceases  to  become  a  solid.  This  concentration 
of  zero  friction  was  found  for  a  certain  English  china  clay  to  be 
19.5  per  cent  by  volume  when  suspended  in  water  containing 
one-tenth  of  1  per  cent  of  potassium  carbonate.  If  the  particles 
of  clay  were  spheres  of  uniform  size,  suspensions  of  this  material 
would  show  plasticity  in  concentrations  of  solid  from  19.5  to 
47.64,  i.e.,  over  a  range  of  roughly  30  per  cent.  Colloidal 
graphite  exhibits  zero  fluidity  when  there  is  only  5.4  per  cent 
in  suspension,  hence  it  has  a  plasticity  range  of  concentrations 
of  over  40  per  cent.  On  the  other  hand,  suspensions  of  many 
coarse  materials  have  a  plasticity  range  which  is  much  con- 
stricted, which  for  practical  purposes,  is  sometimes  a  serious 
disadvantage. 

There  is  abundant  evidence  that  as  the  diameter  of  the 
particles  is  decreased,  the  opportunity  for  the  particles  touching 


230  FLUIDITY  AND  PLASTICITY 

is  increased,  which  enhances  the  friction,  but  this  effect  reaches  a 
limit  eventually  when  the  particles  are  so  small  that  their 
Brownian  movement  becomes  appreciable  and  strains  in  the 
material  are  not  permanent. 

If,  as  we  have  intimated,  the  friction  is  subject  to  the  laws 
of  ordinary  external  friction,  the  friction  should  be  closely 
dependent  upon  the  adhesion  of  the  particles  to  each  other 
but  independent  upon  the  nature  of  the  medium  so  long  as  it  is 
inert.  In  confirmation  of  this  we  note  that  whereas  the  china 
clay  referred  to  above  showed  zero  friction  when  the  volume 
concentration  was  19.5  per  cent,  the  same  clay  thoroughly  shaken 
down  in  a  measuring  flask  in  the  dry  state  showed  a  pore  space  of 
18.4  per  cent,  the  pore  space  in  this  case  being  filled  with  air. 
The  two  values  are  in  very  close  agreement.  Infusorial  earth 
exhibited  zero  fluidity  in  water  when  present  to  the  extent  of 
12.9  per  cent  by  volume,  whereas  in  ethyl  alcohol  the  corre- 
sponding concentration  was  12.1  per  cent.  Finally  it  has  been 
observed  that  the  temperature  and  therefore  the  fluidity  of  the 
medium  is  without  effect  upon  the  friction. 

Adhesion  between  the  particles  may  be  influenced  in  a  marked 
degree  by  the  addition  of  small  amounts  of  substances  of  the 
most  diverse  character.  Generally  speaking,  substances  which 
yield  hydrogen  ions  increase  the  adhesion,  i.e.,  promote  floccula- 
tion,  while  substances  which  yield  hydroxyl  ions  decrease  the 
adhesion  and  promote  deflocculation.  Colloids  also  have  a 
noteworthy  effect.  In  flocculation,  structure  is  produced  and 
therefore  the  friction  is  enhanced.  In  a  given  instance,  using 
50  per  cent  china  clay  in  water,  the  friction  was  lowered  from 
78  to  59.5  by  adding  merely  one-tenth  of  1  per  cent  of  potassium 
carbonate,  which  of  course  yields  hydroxyl  ions. 

The  mobility  is  dependent  upon  the  fluidity  of  the  medium. 
This  in  turn  is  influenced  by  the  temperature,  hence  we  may 
expect  that  the  mobility  of  a  solid  will  be  dependent  upon  the 
temperature.     Thus  in  a  50  per  cent  clay  suspension  the  mobility 
at  25°  was  found  to  be  5.11  and  at  40°,  7.88.     The  ratio  between 
these  mobilities  is  1.54  which  is  very  close  to  the  ratio  of  the 
fluidities  of  water  at  these  two  temperatures 
^  _  166.9  _ 
5°  ~  111.7  ~ 


THE  PLASTICITY  OF  SOLIDS  231 

The  result  of  deflocculation  is  to  greatly  increase  the  mobility. 
Thus  one-tenth  of  1  per  cent  of  potassium  carbonate  raised  the 
mobility  from  1.17  to  5.11  which  is  an  increase  of  over  330  per 
cent,  a  truly  remarkable  effect. 

SEEPAGE  AND  SLIPPAGE 

When  the  shearing  force  is  just  a  little  less  than  the  friction, 
there  is  generally  a  certain  amount  of  flow  which  is  due  to  two 
different  causes.  In  the  first  place,  under  ordinary  conditions 
of  flow  the  pressure  tends  to  cause  the  medium  to  seep  through 
the  material.  With  this  filtration  phenomenon  there  is  a  local 
change  in  concentration  and  therefore  a  change  in  the  char- 
acter of  the  flow.  Seepage  is  unimportant  when  the  medium  is 
viscous  arid  the  suspended  particles  are  small  as  in  paint. 

The  second  difficulty  is  due  to  slippage,  which  comes  from  lack 
of  sufficient  adhesion  between  the  material  and  the  shearing 
surface.  The  shearing  surface  is  wet  with  the  liquid  medium  and 
the  smooth  surface  affords  little  opportunity  for  the  attachment 
or  interlocking  of  the  particles.  The  result  is  that  there  is  a 
layer  of  liquid  between  the  shearing  surface  and  the  main  body  of 
the  suspension  and  flow  takes  place  in  this  layer  according  to 
the  laws  of  viscous  rather  than  plastic  flow.  Green  (1920) 
has  observed  this  phenomenon  in  paint  under  the  microscope, 
the  material  moving  as  a  solid  rod  until  the  shear  reaches  a 
certain  value  when  it  begins  to  move  in  telescoping  layers. 
This  slippage  causes  the  rate  of  flow-shear  curve  to  be  no  longer 
linear  when  the  rate  of  flow  is  small  and  the  curve  passes  through 
the  origin. 

Difficulties  due  to  seepage  and  slippage  can  be  overcome 
by  using  sufficiently  high  pressures,  so  that  the  viscous  flow 
factor  will  become  negligible.  In  this  case  there  should  be  a 
linear  relation  between  shear  and  rate  of  flow. 

HYDRAULIC  FLOW  AND  THE  PLASTIC  STATE 

So  far  as  known  to  the  author,  no  one  has  yet  used  rates 
of  flow  high  enough  to  bring  about  eddy  currents,  which  are  so 
troublesome  in  the  case  of  liquids.  But  there  is  the  same 
necessity  for  using  long  narrow  tubes  for  measuring  the  flow, 


232 


FLUIDITY  AND  PLASTICITY 


rather  than  orifices  or  very  short  tubes,  for  the  flow  of  a  plastic 
material  through  an  orifice  gives  no  idea  of  the  mobility  of  the 
material,  just  as  the  flow  of  a  liquid  through  an  orifice  is  largely 
independent  of  the  viscosity  of  the  liquid.  Flow  through  an 


=  10 

=  11 

•  =  12 

•  =  13 

'  =  14 
=  15 
=  16 


7 


900 


7 


/ 


7 


7 


7 


"0  10  20  JO  40  50  60  70          80          90.'         100        110 

FIG.  81. — Hydraulic  flow  of  a  plastic  material  after  experiments  of  Simonis. 

orifice  does,  however,  lead  to  a  knowledge  of  the  friction  constant 
of  the  plastic  substance,  as  proved  by  the  experiments  of  Simonis 
(1905). 

Simonis  used  40  g  of  Zettlitz  earth    with    100  g  of    water 
to  which  were  added  successive  portions  of  a  dilute  solution  of 


THE  PLASTICITY  OF  SOLIDS  233 

sodium  hydroxide  containing  1.795  g  per  liter.  The  pressure 
seems  to  have  been  measured  as  centimeters  of  water  head,  and 
the  volume  of  flow  in  milliliters  per  600  sec.  He  measured  the 
flow  of  16  mixtures  and  pure  water,  designated  by  the  numbers  on 
the  curves  in  Fig.  81.  The  amounts  of  sodium  hydroxide  solu- 
tion added  are  noted  in  the  second  column  of  Table  LXI. 

The  curves  are  nearly  linear  except  when  the  volume  of  flow 
is  small.  The  curvature  is  probably  due  to  seepage.  '  The  hori- 
zontal distance  of  the  different  curves  from  the  curve  No.  10 
is  evidently  a  relative  measure  of  the  friction  constant.  The 
values  of  the  friction  constant  /  as  obtained  graphically  are  given 
in  the  table.  We  have  found  that  it  is  possible  to  calculate  this 
relative  friction  constant  /'  by  means  of  the  formula 

/'  -  154  -  14. Ic  (94) 

where  c  represents  the  number  of  milliters  of  sodium  hydroxide 
added.  It  appears,  therefore,  from  a  comparison  of 'the  values  of 
/  and  /'  that  Simonis'  experiments  confirm  our  conclusion  that 
the  friction  is  a  linear  function  of  the  concentration.  We  note 
that  the  friction  constant  continually  decreases  as  water  is  added 
until  11  ml  have  been  added  after  which  further  additions  are 
without  effect  upon  the  rate  of  flow.  On  adding  11  ml,  the 
material  reaches  the  concentration  of  zero  fluidity  or  zero  friction, 
and  the  curve  10  should  pass  through  the  origin.  That  the  curves 
10  to  17  all  coincide  with  curve  10  accords  with  what  we  should 
expect  of  liquids  flowing  through  an  orifice. 

The  fact  that  all  of  the  curves  are  sensibly  parallel  constitutes 
the  remarkable  difference  between  flow  through  a  capillary  and 
flow  through  an  orifice.  It  does  not  signify  that  the  plastic 
mixtures  all  have  the  same  mobility  any  more  than  it  signifies 
that  all  of  the  liquid  mixtures  have  the  same  fluidity.  It  means 
simply  that  the  rate  of  flow  through  an  orifice  is  independent 
of  the  fluidity  or  mobility.  If  in  the  equations  for  the  flow  of  a 
viscous  or  a  plastic  substance  through  a  capillary  we  make  the 
length  of  the  capillary  zero,  we  obtain  the  identical  equation 
V 


<mp 

where  A;  is  a  constant.     This  is  the  characteristic  and  familiar 
equation  for  the  flow  of  liquids  through  an  orifice  in  which  the 


FLUIDITY  AND  PLASTICITY 


^«  :::::::  .-ooooooooooooooo 


THE  PLASTICITY  OF  SOLIDS  235 

fluidity  or  mobility  does  not  appear.  The  table  shows  that  the 
higher  rates  of  flow  may  be  calculated  quite  accurately  by  means 
of  the  formula 

^  =  23.4(P  -  /')  +  168.7  (96) 

HISTORICAL 

A  large  amount  of  work  has  been  devoted  to  the  flow  of  solids. 
Methods  of  measuring  plasticity,  consistency,  and  hardness 
have  aimed  to  give  a  single  numerical  value  to  a  property  which 
is  found  to  be  complex.  Plasticity  itself  has  hardly  been  meas- 
ured, but  rather  some  property  instead  which  is  supposed  to  be 
related  to  it,  such  as  the  amount  of  water  required  to  bring  a 
clay  to  a  given  consistency,  the  tensile  strength  on  drying,  the 
absorptive  capacity  for  certain  dyes  such  as  malachite  green,  the 
amount  of  shrinkage  on  drying,  etc.  It  is  no  doubt  true  that 
these  properties  are  dependent  in  large  measure  upon  the  fineness 
of  grain  which  also  essentially  affects  the  plasticity,  but  a  knowl- 
edge of  these  properties  leaves  the  subject  of  plastic  flow  in  a 
nebulous  state. 

Many  investigators  have  investigated  the  so-called  "viscosity 
of  solids,"  assuming  that  solids  obey  the  ordinary  laws  of  viscous 
flow,  and  Tammann  has  identified  fluidity  with  plasticity.  Heyd- 
weiller  (1897)  has  measured  the  viscosity  of  menthol  in  both  the 
liquid  and  the  solid  condition.  Weinberg  (1913)  Dudetzkii 
(1914)  and  Pochettino  (1914)  have  measured  the  viscosity  of 
pitch  or  asphalt.  Segel  (1903)  worked  with  sealing-wax  and 
Barus  (1893)  with  marine  glue.  Barus  made  the  important 
observation  that  if  the  rod  of  material  coming  out  of  the  capillary 
used  in  his  measurements  was  cut  off  neatly  with  a  knife,  the 
cylinders  thus  formed  were  in  a  strained  condition.  They 
spontaneously  change  their  shape,  the  advancing  end  becoming 
hollowed  in  and  the  following  end  being  bulged  out.  This  proof 
of  strain  is  very  similar  to  that  observed  by  Trouton. 

Tresca  (1868)  did  valuable  work  in  forcing  metals  through 
orifices  and  proving  that  they  may  be  made  to  flow  in  a  linear 
manner  much  as  liquids  do.  It  gives  good  reason  for  the  pre- 
sumption that  it  is  practicable  to  measure  the  friction  and  mobil- 


236 


FLUIDITY  AND  PLASTICITY 


ity  of  metals  and  alloys.     The  work  of  Andrade  on  the  differ- 
ent types  of  flow  in  metals  may  be  referred  to. 

Werigen,  Lewkojeff,  and  Tammann  (1903)  measured  the  rate 
of  outflow  of  various  metals  and  arranged  the  metals  in  a  plastic- 
ity series  as  follows:  potassium,  sodium,  lead,  thallium,  tin, 
bismuth,  cadmium,  zinc,  antimony.  They  observed  that  with 
equal  pressures  and  openings,  the  efflux  increases  by  about  100 
per  cent  for  every  rise  of  10°  in  temperature.  This  is  shown  by 
the  following  table: 

TABLE  LXII. — THE  RELATIVE  EFFLUX  OF  METALLIC  LEAD  THROUGH  A  SMALL 

ORIFICE  AT  VARIOUS  TEMPERATURES   (AFTER  WERIGEN,  LEWKOJEFF 

AND  TAMMANN) 


Temperature, 
degrees 

Efflux  (relative) 

Temperature, 
degrees 

Efflux  (relative) 

0.5 

0.8 

60.3 

42.4 

10.4 

1.2 

70.0  . 

84.3 

20.5 

2.3 

79.3 

157.5 

30.4 

4.7 

89.6 

211.5 

50.7 

22.9 

When  a  wire,  which  is  stretched  by  a  weight,  is  subjected 
to  torsional  vibrations,  the  amplitudes  of  the  vibrations  form  a 
series  in  geometrical  progression,  and  therefore  the  logarithmic 
decrement  of  the  amplitude  is  a  constant.  A  part  of  the  energy 
of  vibration  is  given  to  the  surrounding  atmosphere  and  a  part 
is  transmitted  to  the  support,  but  a  portion  of  the  energy  is 
dissipated  within  the  wire  itself.  It  is  generally  agreed  that  this 
loss  is  due  to  the  lack  of  perfect  elasticity  in  the  wire.  In  other 
words,  the  wire  when  subjected  to  shearing  stress  suffers  per- 
manent deformation  even  though  the  stress  is  not  equal  to  the 
elastic  limit.  This  deformation  causes  a  shift  in  the  position 
of  rest,  so  that  as  the  pendulum  passes  from  its  new  position  of 
rest  to  its  old  position  of  rest,  it  does  so  at  the  expense  of  its  own 
momentum  and  there  is  thus  a  loss  of  energy.  This  flow  is 
entirely  analogous  to  the  flow  of  various  plastic  materials  such 
as  clay  slip  and  paint,  which  we  have  already  considered,  when 
the  shearing  stress  is  less  than  the  friction. 


THE  PLASTICITY  OF  SOLIDS  237 

Since  the  flow  is  of  the  nature  of  local  slippage  rather  than  true 
plastic  flow,  strains  accumulate  and  they  remain  after  the  stress 
is  removed.  The  result  is  the  same  as  that  observed  by  Trouton 
in  pitch,  in  that  the  substance  tends  to  creep  slowly  back  toward 
its  old  position  of  rest  during  a  period  of  time  which  in  pitch  is 
comparatively  short  but  may  be  observed  in  metals  for  hours  or 
even  days.  The  elastic  "after  effect"  has  been  the  subject  of 
exhaustive  investigation  by  Weber  (1835),  Warburg  (1869), 
Kohlrausch  (1863-76),  Boltzmann  (1876),  G.  Wiedemann  (1879), 
Pisati  (1879),  Streintz  (1879),  Rakkuk  (1888),  Wiechert  (1889) 
and  others. 

Kupffer  (1860)  was  inclined  to  attribute  this  partial  flow  of 
the  metal  to  what  he  would  denominate  the  fluidity  of  solids 
in  analogy  to  the  fluidity  of  liquids.  He  says,  "II  parait  que 
les  molecules  des  corps  solides  possedent  la  propriete  non  seule- 
ment  de  s'ecarter  les  unes  des  autres  en  produisant  une  resistance 
proportionelle  aux  ecarts,  mais  aussi  de  glisser  les  unes  sur  les 
autres,  sans  produire  aucune  effort.  Cette  propriete  est  possedee 
a  un  haut  degre  par  les  fluides;  je  le  nommerais  volontiers  la 
fluidite  des  corps  solides;  le  coefficient  ty  pourrait  etre  appele 
coefficient  de  fluidite;  la  malleabilite  des  metaux  parait  en  de- 
pendre  et  peut-etre  aussi  leur  durete."  According  to  the  present 
views  we  would  say  that  this  partial  flow  was  evidence  of  low 
friction  or  high  mobility. 

In  harmony  with  this  view,  it  has  been  found  that  the  logarith- 
mic decrement  of  the  amplitude  of  vibration  is  low  in  hard  metals 
like  steel  and  high  in  soft  metals  like  lead.  The  logarithmic 
decrement  also  increases  as  the  temperature  is  raised  but  in  this 
respect  iron  and  steel  are  exceptional  below  100°C  according  to 
Kupffer,  Pisati,  and  Horton  (1905).  It  will  be  recalled  that 
sulfur  presents  a  similar  exception  in  the  case  of  liquids. 

According  to  this  view,  the  elastic  limit  is  reached  when  the 
shearing  stress  is  equal  to  the  friction  constant,  for  at  this  value 
of  the  stress  the  material  begins  to  yield.  But  since  the  deforma- 
tion takes  place  with  exceeding  slowness  at  this  particular  stress, 
a  wire  may  be  loaded  considerably  beyond  the  elastic  limit  before 
the  flow  becomes  appreciable.  The  yield  point  naturally  depends 
to  some  extent  upon  the  rate  with  which  the  load  is  put  on.  • 

Just  as  Trouton  found  that  a  given  shearing  stress  produced  a 


238  FLUIDITY  AND  PLASTICITY 

more  rapid  rate  of  flow  at  first  than  later  when  the  strains  were 
developed  to  their  maximum  amount,  so  it  is  common  experience 
that  metals  become  harder  with  working,  but  that  they  may  be 
softened  again  by  annealing.  In  the  process  of  annealing,  the 
plasticity  is  increased  by  raising  the  temperature  and  thus  the 
strains  relieve  themselves  more  quickly  than  otherwise  would  be 
the  case. 

An  entirely  different  view  from  that  given  above  has  been 
presented  by  Lord  Kelvin  and  it  has  had  many  followers.  Noting 
that  the  logarithmic  decrement  of  the  vibration  is  greater  in  lead 
and  zinc  than  it  is  in  steel,  he  reasoned  as  follows: 

"Hence,  there  is  in  elastic  solids  a  molecular  friction  which  may  be 
properly  called  viscosity  of  solids,  because  as  being  an  internal  resistance 
to  change  of  shape  depending  on  the  rapidity  of  the  change,  it  must  be 
classed  with  fluid  molecular  friction,  which  by  general  consent  is  called 
viscosity  of  fluids.'1 

However,  he  further  stated: 

"  But  at  the  same  time  it  ought  to  be  remarked  that  the  word  viscosity, 
as  used  hitherto  by  the  best  writers,  when  solids  or  heterogeneous  semi- 
solid-semi-fluid  masses  are  referred  to,  has  not  been  distinctly  applied  to 
molecular  friction,  especially  not  to  molecular  friction  of  a  highly  elastic 
solid  within  its  limits  of  high  elasticity,  but  has  rather  been  employed  to 
designate  a  property  of  slow  continual  yielding  through  very  great,  or 
altogether  unlimited,  extent  of  change  of  shape,  under  the  action  of 
continued  stress." 

It  has  thus  come  about  that  the  logarithmic  decrement  has 
been  taken  as  a  measure  of  the  viscosity  of  a  metal,  so  that 
according  to  this  nomenclature  lead  has  a  higher  viscosity  than 
steel  and  the  viscosity  of  lead  increases  as  the  temperature  is 
raised,  which  point  of  view  is  just  the  opposite  of  that  used 
by  Kupffer  and  to  which  we  are  generally  familiar  in  discussing 
the  viscosity  of  fluids.  Since,  however,  several  investigators  have 
followed  Lord  Kelvin  in  his  nomenclature,  there  is  danger  of 
considerable  confusion.  If  we  hereafter  refer  to  the  friction  and 
mobility  of  solids,  the  term  "viscosity  of  solids"  becomes 
unnecessary;  and  we  may  confidently  expect  that  the  friction 
constant  of  lead  will  be  found  to  be  lower  than  that  of  steel  and 
that  it  will  decrease  with  the  temperature. 


THE  PLASTICITY  OF  SOLIDS  239 

In  conclusion,  we  note  again,  cf.  page  58,  that  Reiger  (1906) 
and  Glaser  (1907)  have  carefully  investigated  the  question  as  to 
whether  the  laws  of  Poiseuille  may  be  applied  to  soft  solids,  using 
as  their  material  suspensions  of  colophony  in  turpentine.  They 
concluded  that  with  a  tube  having  a  radius  of  0.49  cm  the  vis- 
cosity was  independent  of  the  pressure  between  the  limits  of 
136  and  2,172  g  per  square  centimeter;  and  in  a  similar  way  it 
was  independent  of  the  length  of  the  tube  for  lengths  varying 
between  2.4  and  20.6  cm.  They  found  that  with  a  pressure  of 
1,965  g  per  square  centimeter,  if  they  varied  the  radius  of  the 
tube  from  1.52  to  0.34  cm,  the  viscosity  remained  constant  but 
for  tubes  of  smaller  radii  the  viscosity  rapidly  increased  until 
finally  the  material  seemed  to  have  infinite  viscosity.  This 
inferior  limit  is  unlike  anything  observed  in  the  flow  of  liquids,  for 
the  smaller  the  radius  of  the  tube,  the  better  are  the  laws  of 
Poiseuille  obeyed,  and  in  large  tubes  the  flow  is  largely  inde- 
pendent of  the  viscosity  of  the  fluid.  It  seems  probable  that  the 
use  of  such  very  large  tubes  has  prevented  Reiger  and  Glaser 
from  discovering  the  friction  constant  just  as,  in  the  period  before 
Poiseuille's  study  of  flow  in  capillaries,  the  use  of  large  tubes 
prevented  the  discovery  of  the  laws  of  viscous  flow.  In  large 
tubes  the  shearing  stress  is  very  large  in  comparison  with  the 
friction  which  may  possibly  explain  the  fact  that  the  "viscosity" 
was  found  to  be  independent  of  the  pressure  or  length  of  the  tube. 

We  note  that  the  inferior  limit  of  the  radius  of  the  tube  is 
increased  as  the  percentage  of  solid  in  the  mixture  is  increased. 
This  is  what  we  should  expect  since  this  procedure  raises  the 
friction  constant.  With  an  80  per  cent  of  colophony  the  lower 
limit  of  the  radius  was  found  to  be  0.100  cm,  with  an  85  per  cent 
mixture  it  was  0.576  cm.,  and  with  a  90  per  cent  mixture  it  was 
1.019  cm.  We  give  below  a  re'sume'  of  the  data  of  Glaser  for  the 
90  per  cent  suspension  of  colophony  in  turpentine,  the  pressure 
throughout  being  2,040  g.  per  square  centimeter. 

The  subject  of  the  plasticity  of  ice  takes  on  exceptional 
interest  and  importance  in  connection  with  the  flow  of  glaciers 
and  it  has  been  the  object  of  research  by  many  investigators, 
among  whom  we  mention  Pfaff  (1875),  McConnel  (1886), 
Miigge  (1895),  Hess  (1902),  Weinberg  (1905)  and  Deeley  and 
Parr  (1914).  It  is  a  noteworthy  fact  that  the  precipitous  moun- 


240 


FLUIDITY  AND  PLASTICITY 


TABLE  LXIII. — THE  EFFECT  OF  VARYING  THE  RADIUS  OF  THE  CAPILLARY 
ON  THE  "VISCOSITY  OF  A  SOLID"  (AFTER  GLASER) 


Temperature, 

Radius, 

Length, 

Time, 

Volume, 

"Viscosity," 

degrees    C 

centimeters 

centimeters 

seconds 

centimeters 

absolute 

12.2 

1.525 

25.1 

16,200 

0.331 

4.  59X10* 

12.3 

1.241 

15.9 

43,200 

11.20 

4.54X10" 

12.3 

1.019 

15.9 

173,000 

2.060 

4.59X108 

12.3 

0.746 

16.0 

258,000 

0.756 

5.62X10» 

12.3 

0.576 

15.1 

171,000 

0.129 

7.91X10' 

12.3 

0.364 

15.8 

350,000 

0.0866 

25.2X10 

tain  peaks  maintain  their  sharp  outlines  through  geological  ages 
whereas  ice  flows  steadily  in  spite  of  apparent  hardness.  This 
indicates  that  the  friction  constant  of  ice  is  incomparably  lower 
than  that  of  most  silicate  rocks.  Whereas  the  glacier  scrapes  its 
bed  to  some  extent  (slipping),  there  is  an  abundance  of  evidence 
that  there  is  differential  flow  in  the  glacier  mass,  so  that  although 
regelation  introduces  a  new  factor  into  the  problem,  the  flow  is 
essentially  plastic  in  its  nature. 


CHAPTER  IX 
THE  VISCOSITY  OF  GASES 

In  1846,  the  same  year  in  which  Poiseuille  published  his 
principal  paper  on  the  laws  of  viscous  flow  in  liquids,  Thomas 
Graham  published  the  first  of  a  series  of  papers  on  the  "trans- 
piration" of  gases  through  tubes  of  small  diameter,  which 
have  great  historic  interest.  Graham  sharply  differentiated  the 
flow  of  gases  through  an  aperture  (effusion)  and  flow  through  a 
long  narrow  tube  (transpiration) ;  he  noted  that  the  resistance  of 
a  tube  of  a  given  diameter  was  directly  proportional  to  its 
length.  Also  "  dense  cold  air  is  transpired  most  rapidly,"  and  his 
experiments  led  him  to  a  relation  between  the  time  of  transpira- 
tion and  the  density  of  the  gas.  Graham  studied  the  effect  of 
different  pressures  and  concluded  that  "for  equal  volumes  of  air 
of  different  densities,  the  times  of  transpiration  are  inversely 
as  the  densities,"  as  exemplified  in  the  following  table: 

TABLE   LXIV. — THE    EFFECT   OF    PRESSURE    UPON   THE  TRANSPIRATION 
OF  AIR  (FROM  GRAHAM) 


Pressure,  atmospheres 

Observed  time  ol  trans- 
piration for  equal  vol- 
umes (relative) 

Calculated  time 

1.0 

1.0 

1.0 

1.25 

0.795 

0.800 

1.5 

0.673 

0.667 

1.75 

0.589 

0.571 

2.0 

0.524 

0.5 

When  Clausius  proposed  the  kinetic  theory  in  1857,  all  of 
the  properties  of  gases  took  on  increased  interest,  and  Maxwell 
in  1861  published  a  paper  in  which  he  discussed  the  three  kinds 
of  diffusion:  (1)  Diffusion  of  heat  or  conductivity,  (2)  Diffusion  of 
matter,  and  (3)  Diffusion  of  motion  or  viscosity.  The  third  or 
16  241 


242  FLUIDITY  AND  PLASTICITY 

viscosity  is  the  simplest  to  obtain  and  it  may  be  used  to  calculate 
the  other  two,  so  viscosity  played  an  exceedingly  important  part 
in  the  years  that  followed  in  the  establishment  of  the  kinetic 
theory  on  a  firm  basis.  Maxwell  defined  the  unit  of  viscosity; 
and  the  theory  of  viscosity  and  its  measurement  was  rapidly 
advanced  by  Maxwell,  O.  E.  Meyer  and  many  others.  After 
many  vicissitudes,  the  conclusion  was  reached  that  viscosity 
is  a  fundamental  property  independent  of  the  particular  method 
used  in  its  measurement.  Thus,  for  instance,  Millikan  (1913) 
brought  together  the  results  for  air  at  23°  by  five  different 
methods  and  found  them  to  agree  to  within  less  than  0.1  per  cent 
as  given  in  Table  LXV. 

TABLE  LXV. — THE  VISCOSITY  OP  AIR  AT  23°C  (FROM  MILLIKAN) 
0.00018258     Tomlinson Damping  of  the  swinging  of  a 

pendulum (1886) 

0.00018229     Hogg Damping     of     an     oscillating 

cylinder (1905) 

0. 00018232     Grindley  and  Gibson .  Flow  through  a  large  tube (1908) 

0.00018257     Gilchrist Method  of  constant  deviation. .    (1913) 

0. 00018227     Rapp Transpiration  method (1913) 


0 . 00018240     Average  value 

Between  12  and  30°  the  viscosity  of  air  is  given  by  the  following 
formula  with  an  accuracy  of  nearly  0.1  per  cent  according  to 
Millikan : 

r,t  =  0.00018240  -  0.000000493  (23°  -  0 

The  reader  may,  however,  be  referred  to  the  more  recent  paper  of 
Vogel  (1914). 

THE  THEORY  OF  THE  VISCOSITY  OF  GASES 

The  theory  of  gaseous  viscosity  has  been  so  often  stated 
that  it  need  be  stated  here  only  in  the  simplest  terms.  The 
viscosity  of  a  gas  is  given  by  the  tangential  force  required  per 
unit  area  to  maintain  a  unit  velocity  in  a  plane  of  indefinite 
extent  at  a  unit  distance  from  another  parallel  plane  supposed 
to  be  at  rest,  the  space  between  the  planes  being  occupied  by  the 
gas.  It  is  assumed  that  if  the  shearing  force  is  equal  to  the  vis- 
cosity, the  velocity  v  at  any  point  will  be  numerically  equal  to  its 


THE  VISCOSITY  OF  GASES  243 

distance  s  from  the  plane  which  is  at  rest.  If,  with  Joule,  we 
think  of  one-third  of  the  molecules  as  moving  in  a  direction  which 
is  at  right  angles  to  the  shear,  then  these  molecules  are  the  only 
ones  concerned  in  the  transfer  of  momentum  which  is  the  cause 
of  viscosity  in  gases.  Through  a  unit  area  of  a  plane  separating 
any  two  layers  of  fluid  there  will  pass  per  second  in  either  direc- 
tion 1/QNV  molecules,  N  being  the  number  of  molecules  in  a 
unit  volume  and  V  their  average  velocity  as  calculated  from  the 
kinetic  energy.  The  molecule  in  passing  through  the  given  plane 
comes  from  a  distance  which  is  equal  to  the  molecular  mean  free 
path  L,  and  therefore  from  a  plane  in  which  the  velocity  is  not  v 
but  v  —  L  in  one  direction  and  v  -\-  L  in  the  other  direction. 
The  molecule  which  diffuses  into  a  more  slowly  moving  layer 
loses  momentum  represented  by  m(v  —  L),  and  similarly  a 
molecule  diffusing  into  the  more  rapidly  moving  layer  gains 
momentum  represented  by  m(v  +  L),  so  that  the  total  loss 
of  momentum  is 

r,=±NVm[(v-L)  -(v+L)] 


or  since  Nm  =  p 

1  =  ^pVL  (97) 

If  the  speed  of  the  molecules  fi  is  the  mean  value  as  calculated 
according  to  Maxwell's  law  of  distribution,  the  formula  for  the 
viscosity  becomes,  according  to  0.  E.  Meyer  (1889), 

rj  =  0.  30967  ftL  (98) 

Since  the  length  of  the  mean  free  path  varies  inversely  as  the 
pressure,  whereas  the  density  varies  directly  as  the  pressure, 
it  was  seen  at  once  that  the  viscosity  of  gases  should  be  inde- 
pendent of  the  pressure.  This  surprising  result  was  confirmed 
by  O.  E.  Meyer  (1866)  calculating  out  the  measurements  of 
Graham,  also  by  the  measurements  of  Maxwell  (1866)  and  O.  E. 
Meyer  (1865),  and  it  did  much  to  establish  the  kinetic  theory. 
With  the  acceptance  of  the  kinetic  theory  it  can  be  seen  that  vis- 
cosity measurements  give  a  very  convenient  and  simple  method 
for  the  determination  of  the  mean  free  path. 


244  FLUIDITY  AND  PLASTICITY 

TABLE  LXVI. — EVIDENCE  FROM   MAXWELL   (1866)   THAT  THE  VISCOSITY 
OF   AIR   is   INDEPENDENT   OF   THE    PRESSURE 


Temperature,  degrees 
Centigrade 

Pressure  in  mercurial 
inches 

Logarithmic  decrement 
of    oscillating    disks 

12.8 
12.8 
13.3 

0.50 
5.52 

29.90 

0.15378 
0.15379 
0.15398 

Warburg  and  Babo'  (1882)  were  the  first  to  prove  that  the 
viscosity  of  a  gas  fluctuates  widely  with  the  pressure  in  the 
neighborhood  of  the  critical  temperature,  using  carbon  dioxide 
as  their  experimental  substance.  We  have  already  commented 
upon  the  data  for  this  substance  recently  obtained  by  Phillips. 

Kundt  and  Warburg  (1875)  measured  the  viscosity  of  carbon 
dioxide  by  the  disk  method  at  pressures  as  low  as  0.1  mm  of 
mercury  and  they  found  that  the  logarithmic  decrement  of  the 
amplitude  of  vibrations  became  noticeably  smaller  when  the 
pressure  became  less  than  about  1.5  mm,  the  distance  between 
the  disks  being  from  1  to  3  mm.  At  atmospheric  pressure  the 
molecular  mean  free  path  of  carbon  dioxide  at  0°  is  0.0000065 
cm,  and  at  2  mm  the  mean  free  path  is  therefore  approximately 
0.02  mm.  Since  a  considerable  portion  of  the  molecules  depart 
widely  from  the  mean  velocity,  we  should  expect  the  viscosity  to 
decrease  long  before  the  molecular  mean  free  path  became  equal 
to  the  distance  between  the  boundary  surfaces.  Kundt  and 
Warburg  believed  that  the  decrease  in  viscosity  due  to  the  in- 
creasing length  of  the  mean  free  path  should  not  occur  so  long  as 
the  thickness  of  gas  was  14  times  the  mean  free  path  and  they 
therefore  assumed  that  at  high  exhaustions  there  is  "slipping" 
at  the  boundary.  No  one  has  yet  explained  why  a  molecule  of  a 
rarefied  gas  is  any  less  likely  to  give  up  its  translational  velocity 
than  a  molecule  of  gas  at  ordinary  pressures.  Whether  the 
decrease  in  the  viscosity  is  due  to  the  increase  in  the  free  path  or 
not,  the  hypothesis  of  slipping  seems  improbable,  and  there  may 
be  some  other  explanation  for  the  results  observed.  For  example, 
in  the  case  of  the  experiments  of  Kundt  and  Warburg  with  hydro- 
gen, the  decrease  in  viscosity  at  moderately  low  pressures  is, 
according  to  Crookes,  "  probably  due  to  the  presence  of  a  trace  of 


THE  VISCOSITY  OF  GASES  245 

foreign  gas,  most  likely  water,"  which  seems  to  have  been  sus- 
pected by  Kundt  and  Warburg  themselves. 

Crookes  (1881)  measured  the  logarithmic  decrement  of 
a  mica  disk  swinging  in  a  glass  bulb  and  supported  by  a  glass 
fiber,  using  pressures  as  low  as  could  be  measured,  by  means  of  a 
McLeod  gage.  The  gases  employed  were  air,  oxygen,  nitrogen, 
carbon  dioxide,  carbon  monoxide,  and  hydrogen  at  15°C.  In 
the  case  of  hydrogen  the  logarithmic  decrement  was  found  to  be 
almost  perfectly  constant  from  atmospheric  pressure  down  to 
0.25  mm.  At  about  this  pressure  the  viscosity  of  all  gases 
decreases  rather  suddenly.  With  other  gases  there  is  a  slow 
decrease  with  the  pressure  even  from  atmospheric  pressure, 
except  in  a  sample  of  air  which  contained  some  water  vapor, 
in  which  case  the  logarithmic  decrement  was  at  first  that  of 
air,  but  at  about  50  mm  it  decreased  rapidly  to  that  of  pure 
hydrogen.  In  an  absolute  vacuum  we  must  assume  that  the 
fluidity  is  infinite,  hence  Maxwell's  law  must  break  down  at  very 
low  pressures. 

According  to  the  data  of  Phillips,  Fig.  54,  we  should  expect 
that  Maxwell's  law  would  break  down  at  low  temperatures 
or  at  very  high  temperatures.  There  is  a  curious  dearth  of  data 
with  which  to  test  out  this  point.  However,  a  hydrocarbon 
vapor,  "  kerosoline, "  was  measured  by  Crookes  and  the  viscosity 
was  found  to  decrease  rapidly  from  the  highest  pressure  obtained 
of  82.5  mm  down  to  8  mm.  Lothar  Meyer  found  in  experiment- 
ing with  benzene  that  the  viscosity  of  the  saturated  vapor  was 
smaller  the  higher  the  back  pressure  at  the  exit  end  of  the  capil- 
lary tube.  At  high  temperatures  we  are  led  to  expect  that  just 
the  opposite  conduct  will  be  observed,  viz.,  that  the  viscosity  will 
decrease  as  the  pressure  is  increased,  see  Fig.  54,  but  there  is  so 
far  as  known  to  the  author  no  data  to  support  this  conclusion. 

VISCOSITY  OF  GASES  AND  TEMPERATURE 

From  the  formula 

•n  =  l/3PVL 

it  is  evident  that  the  effect  of  an  increase  in  temperature  will  be 
to  increase  the  mean  velocity,  but  it  is  not  known  what  effect 
the  temperature  may  have  upon  the  mean  free  path,  although  it 


246  FLUIDITY  AND  PLASTICITY 

seems  most  reasonable  to  assume  that  the  temperature  is  without 
effect,  in  which  case  we  should  expect  the  diffusional  viscosity 
to  vary  directly  as  the  square  root  of  the  absolute  temperature. 
Maxwell  concluded  from  his  experiments  that  the  viscosity 
varies  directly  as  the  first  power  of  the  absolute  temperature. 
Barus  (1889)  worked  with  air  and  with  hydrogen  over  a  very 
wide  range  of  temperature  from  0  to  1,300°  and  found  that  the 
viscosity  increased  as  the  two-thirds  power  of  the  absolute 
temperature.  Holman  (1877)  and  (1886))  in  a  careful  investiga- 
tion of  the  subject  had  found  the  exponent  to  be  0.77  for  air.  On 
the  other  hand,  easily  condensible  gases  and  vapors  such  as 
mercury,  carbon  dioxide,  ethylene,  ethyl  chloride  and  nitrogen 
peroxide  give  values  of  the  exponent  which  are  nearly  unity, 
according  to  Puluj  (1876)  and  Obermayer  (1876);  but  E.  Wiede- 
mann  (1876)  discovered  that  the  value  grows  smaller  as  the  tem- 
perature is  elevated,  which  we  might  have  anticipated  since  they 
thus  become  more  nearly  like  the  permanent  gases.  The  vis- 
cosity of  many  vapors  increases  even  more  rapidly  than  the  first 
power  of  the  temperature.  Schumann  (1884)  used  the  formula 

77  =  KT*.  (99) 

Sutherland  (1893)  believes  that  "the  whole  of  the  discrepancy 
between  theory  and  experiment  will  disappear  if  in  the  theory 
account  is  taken  of  molecular  force.  *  *  Molecular  attraction 
has  been  proved  to  exist,  and,  though  negligible  at  the  average 
distance  apart  of  molecules  in  a  gas,  it  is  not  quite  negligible 
when  two  molecules  are  passing  quite  close  to  one  another;  it 
can  cause  two  molecules  to  collide  which  in  its  absence  might 
have  passed  one  another  without  collision;  and  the  lower  the 
velocities  of  the  molecules,  the  more  effective  does  molecular 
force  become  in  bringing  about  collisions  which  would  be  avoided 
in  its  absence. 

"Molecular  force  alone  without  collisions  will  not  carry  us 
far  in  the  explanation  of  viscosity  of  gases  as  known  to  us  in 
nature,  because  in  all  experiments  on  the  viscosity  of  gases  there 
is  a  solid  body  which  either  communicates  to  the  gas  motion 
parallel  to  its  surface  or  destroys  such  motion,  so  that  the  mole- 
cules of  gas  must  collide  with  the  molecules  of  the  solid ;  for  if  the 
molecules  of  gas  and  solid  act  on  one  another  only  as  centers  of 


THE  VISCOSITY  OF  GASES  247 

force,  then  each  molecule  of  gas  when  it  comes  out  of  the  range 
of  the  molecular  force  of  the  solid  must  have  the  same  kinetic 
energy  as  when  it  went  in,  so  that  without  collision  between 
molecules  of  gas  and  solid  there  can  be  no  communication  of 
motion  to  the  gas.  If,  then,  molecules  of  gas  and  solid  collide, 
molecules  of  gas  must  collide  amongst  themselves." 

In  the  theory  of  diffusional  viscosity  explained  earlier  it  was 
made  plain  that  there  would  be  viscous  resistance  even  if  the 
molecules  failed  to  collide  with  each  other  entirely.  But  Suther- 
land's view  is  in  accordance  with  the  one  we  have  developed  as 
"collisional  viscosity"  in  that  collisions  between  the  molecules, 
whatever  be  the  nature  and  origin  of  the  collisions,  have  an  effect 
upon  the  viscosity.  Sutherland  attributes  the  effect  largely  to 
the  attraction  between  the  molecules,  whereas  the  law  of  Bats- 
chinski  would  lead  us  to  ascribe  the  effect  to  the  volume  of  the 
molecules.  The  two  points  of  view  are  essentially  the  same. 

Sutherland's  theory  led  him  to  the  formula 


77  oc 


or 

1=*^^?  •(«£>)*         aoo) 


where  ot  is  the  coefficient  of  expansion  of  a  gas  and  C  is  a  constant. 
This  formula  has  had  the  most  remarkable  success  of  any  that 
have  been  proposed,  although  it  does  not  apply  to  vapors  well. 
A  single  example  of  its  performance  is  given  in  the  following 
table,  using  Holman's  (1886)  data  for  carbon  dioxide  at  atmos- 
pheric pressure. 

Examining  Sutherland's  formula,  we  observe  that  when  the 
constant  C  is  small  in  comparison  with  the  absolute  temperature 
the  formula  reduces  to  the  simple  theoretical  formula 

7,  =  KT" 

The  discovery,  (cf.  Vogel  (1914)),  that  Sutherland's  formula  fails 
at  low  temperatures  indicates  that  it  does  not  quite  correctly 
take  account  of  the  deviation  from  the  simple  formula. 

Quite  in  harmony  with  the  above,  it  is  found  that  the  values 


248 


FLUIDITY  AND  PLASTICITY 


TABLE    LXVII. — CONCORDANCE    BETWEEN  SUTHERLAND'S  FORMULA  AND 
HOLMAN'S  DATA  FOR  CARBON  DIOXIDE.     C  =  277,  ij0  =  0.000,138,0 


Temperature,  degrees 
Centigrade 

•n  X  107  observed 

77  X  107  calculated 

18.0 

,474 

,471 

41.0 

,581 

,584 

59.0 

,674 

,671 

79.5 

,773 

,766 

100.2 

,864 

,864 

119.4 

,953 

,951 

142.0 

2,048 

2,056 

158.0 

2,121 

2,127 

181.0 

2,234 

2,227 

224.0 

2,411 

2,409 

of  C  for  different  substances  increase  with  the  critical  tempera- 
ture or  boiling-point  of  the  substance.  Rankine  (1910)  obtained 
an  empirical  relation  between  C  and  the  absolute  critical  tempera- 
ture Tcr 

Tcr  =  1.12C  (101) 


TABLE  LXVIII. — THE  RELATION  OF  THE  CONSTANT  C  IN  SUTHERLAND'S 
EQUATION  TO  THE  BOILING-POINT  AND  CRITICAL  TEMPERATURE 


Tcr,  Critical 

Tb,  Boiling 

Substance 

tempera- 
ture, ab- 

C 

Tcr/C 

tempera- 
ture, ab- 

c/n 

solute 

solute 

Helium. 

9  0 

78  2 

0  11 

4  3 

18  3 

Hydrogen 

37  0 

83  0 

0.45 

20  4 

4.1 

Nitrogen  

127.0 

113.0 

.12 

77.5 

.45 

Carbon  monoxide  

133.0 

100.0 

.33 

83.0 

.20 

Oxygen  

154.0 

138.0 

.12 

90.6 

.52 

Nitric  oxide 

179  5 

167.0 

.08 

120  0 

.39 

Ethylene  

383.6 

249.0 

.14 

170.0 

.46 

Carbon  dioxide  

304.0 

259.0 

.17 

194.0 

.33 

Ammonia  

423.0 

352.0 

.20 

240.0 

.47 

Ethyl  ether  

467.0 

325.0 

1.43 

307.0 

1.06 

THE  VISCOSITY  OF  GASES 


249 


which  suggested  to  Vogel  a  similar  relation  to  the  absolute 
boiling  temperature  Tb 

C  =  \AlTb  (102) 

This  formula  indicates  that  C  increases  considerably  more 
rapidly  than  the  temperature,  and  since  Tb  is  comparatively 
large  for  vapors,  the  less  perfect  agreement  of  Sutherland's 
formula  is  partially  explained.  This,  however,  is  not  true  of 
hydrogen  and  helium  which  present  curious  anomalies,  as  shown 
in  Table  LXVIII. 

VISCOSITY  AND  CHEMICAL  COMPOSITION 

If  the  mass  of  a  particle  in  a  rarefied  gas  is  increased  n-fold 
by  changing  its  chemical  composition,  the  velocity  will  be 
n~^  times  the  original  velocity,  so  that  the  momentum  of  each 

TABLE  LXIX. — THE  VISCOSITIES  OF  PERMANENT  GASES  AND 
VAPORS  AT  0°C 


Substance 

Molecular 
weight 

rio  X  107 

Tcr 

7?cr    X    107 

Hydrogen 

2.0 

850 

31.0 

Helium 

4.0 

1,871 

5. 

21 

Methane 

16.0 

1,033 

183. 

Neon  

20.2 

2,981 

Nitrogen  

28.0 

1,678 

Carbon  monoxide 

28.0 

1,672 

133. 

Oxygen  

32.0 

1,'920 

154. 

Argon  

39.9 

2,102 

155.6 

1,253 

Nitrous  oxide  

44.0 

1,362 

Krypton  

82.9 

2,334 

210.5 

1,806 

Xenon  

130.2 

2,107 

288. 

2,266 

Ethyl  alcohol  

46.0 

827 

513. 

Acetone  

58.0 

725 

510. 

Methyl  formate  

60.0 

838 

485. 

Ethyl  ether  

74.1 

689 

467. 

Benzene  

78.0 

689 

561. 

Methyl  isobutyrate  

88.1 

701 

543. 

Ethyl  acetate  

88.1 

690 

523. 

Ethyl  propionate  

90.1 

701 

547'. 

250 


FLUIDITY  AND  PLASTICITY 


molecule  will  be  w^-fold  that  of  the  smaller  molecule.  But  the 
number  of  excursions  of  the  molecules  will  be  in  proportion  to 
n~^,  so  that  the  total  loss  of  momentum  will  be  the  same  as 
before,  provided  only  that  the  number  of  particles  per  unit 
volume  remains  the  same. 

In  gases  at  ordinary  pressure,  there  are  considerable  differences 
in  viscosity  ranging  from  0.0000689  for  benzene  vapor  to 
0.0002981  for  neon,  but  they  are  inconsiderable  as  compared 
with  the  vast  differences  we  find  in  the  liquid  state  and  these 
viscosities  are  measured  at  0°  and  not  under  corresponding  con- 
ditions. Table  LXIX  shows  that  the  vapors  have  viscosities 
which  are  smaller  than  those  of  the  permanent  gases  except 


FIG.  82. — The  relation  between  the  viscosity  of  the  elements  at  their  critical 
temperature  and  their  atomic  weights. 

hydrogen.  Their  viscosities  are  so  nearly  identical  that  it  is  not 
certain  whether  the  viscosity  of  a  given  class  of  chemical  com- 
pounds such  as  the  ethers  differs  from  that  of  the  esters  or 
ketones.  It .  is  quite  impracticable  with  the  data  at  hand  to 
assign  any  effect  to  an  increase  in  the  molecular  weight  within 
a  given  class  of  compounds. 

Since  the  viscosities  of  the  permanent  gases  at  0°  are  not 
simply  related  to  each  other,  it  is  natural  to  seek  some  other 
basis  of  comparison,  and  Rankine  (1911)  has  achieved  success 
along  this  line  by  comparing  the  viscosities  of  the  rare  gases  rjc 
and  their  atomic  .weights  M  at  the  critical  temperatures.  He 
finds  them  related  together  by  the  formula 
r,,*  =  3.93  X  10-10M 


THE  VISCOSITY  OF  GASES  251 

as  depicted  in  Fig.  82.  The  critical  constants  of  neon  and  niton 
have  not  yet  been  determined.  Rankine  has  further  found  that 
the  same  general  formula  applies  to  the  halogens,  but  the  constant 
is  different  being  10.23  X  10~10.  He  gives  for  chlorine  t\c  = 
1,897  X  10-7  and  for  bromine  r,c  =  2,874  X  1Q-7  (cf.  Fig.  82). 
Were  we  to  use  the  molecular  weights  instead  of  the  atomic 
weights,  the  constant  would  be  5.12  X  10~10  which  is  nearer  that 
of  the  rare  gases  but  still  not  identical  with  it. 

THE  VISCOSITY  OF  GASEOUS  MIXTURES 

Since  in  a  rarefied  gas  the  viscosity  is  proportional  to  the 
number  of  molecules  in  a  unit  volume,  i.e.,  to  the  pressure, 
the  viscosities  will  be  additive  when  gases  are  mixed  in  varying 
percentages  by  volume;  but  since  the  viscosity  of  a  rarefied  gas 
is  also  independent  of  the  weight  of  the  molecules,  the  law 
loses  its  significance. 

In  gaseous  mixtures  at  ordinary  pressures  the  simple  deduced 
formula 


still  applies,  it  being  merely  necessary  to  find  the  appropriate 
mean  values  of  p,  V,  and  L.  This  has  been  done  by  Maxwell 
(1868)  and  Puluj  (1879),  and  one  obtains  the  formula  (cf. 
Meyer's  Kinetic  Theory  of  Gases,  page  201  et  seq.) 


.p         p     \r?2/         w2         J 

Graham  (1846)  observed  that  mixtures  of  oxygen  and  nitrogen 
or  oxygen  and  carbon  dioxide  in  all  proportions  have  rates 
of  transpiration  which  are  the  arithmetical  mean  of  the  two 
components.  Thus  for  air, 

0.0001678  X  0.7919  =  0.0001329 
0.0001920  X  0.2081  =  0.0000399 


Calculated  viscosity  of  air 0 . 0001728 

Observed  viscosity  of  air 0. 0001724  Vogel  (1914). 

Graham  and  others  have  noticed  that  when  hydrogen  is  mixed  in 


252  FLUIDITY  AND  PLASTICITY 

small  amounts  with  other  gases,  as  carbon  dioxide  or  methane,  the 
viscosity  of  the  mixture  is  much  greater  than  would  be  calculated 
by  the  simple  formula  of  additive  viscosities.  In  these  cases 
Puluj  (1879)  and  Breitenbach  (1899)  have  found  that  the  more 
complicated  formula  (103)  gives  good  agreement. 

VISCOSITY  OF  GASES  AND  DIFFUSION  AND  HEAT  CONDUCTIVITY 

We  note  that  the  diffusion  coefficient  D  in  a  mixture  of  gases  is 

D  =  1  7i(N2L^,  +  NtLtQj/N  (104) 


NI,  LI,  and  Oi  being  the  number  of  molecules  of  the  first  kind 
of  gas  per  unit  volume,  the  length  of  the  mean  free  path,  and 
the  mean  speed  respectively,  etc.  Also  N  =  NI  +  N%.  Since 
the  length  of  the  mean  free  path  can  most  easily  be  calculated 
from  the  viscosity,  it  becomes  possible  to  calculate  the  diffusion 
coefficient  from  the  viscosity. 

In  the  conduction  of  heat  the  two  kinds  of  gas  become  identical, 
hence  the  above  equation  becomes 

D  =  1  u-flL  (105) 

O 

If  we  neglect  the  small  difference  between  fii  and  ft  due  to 
temperature  difference  the  conductivity  of  heat  k  becomes 

k  =  I  vQLpC,  (106) 

o 

Cv  being  the  specific  heat  of  the  gas  at  constant  volume,  and 
combining  this  equation  with  the  viscosity  Eq.  (97)  we  obtain 
k  =  Cr,Cv  (107) 

C  being  a  constant  (cf.  Eucken  (1913)). 

DETERMINATION  OP  THE  ULTIMATE  ELECTRICAL  CHARGE 

It  is  well  known  that  Sir  J.  J.  Thomson  (1898)  devised  a 
method  for  measuring  the  charge  on  the  particle  of  a  rarefied 
gas  e  by  observing  the  rate  of  fall  under  gravity  of  the  particles  of 
an  ionized  fog  which  had  been  produced  by  sudden  expansion  and 
then  observing  the  rate  of  fall  of  a  similar  cloud  when  it  is  sub- 
jected to  the  action  of  a  vertical  electrical  field  of  known  intensity 
superimposed  upon  gravity. 


THE  VISCOSITY  OF  GASES  253 

If  v  is  the  velocity  of  a  droplet  of  mass  m,  density  p  under  the 
action  of  gravity  alone,  and  v\  its  velocity  when  under  the  in- 
fluence of  the  electrical  field  whose  strength  is  X  in  electrostatic 
units,  then 

i  =  -jy  dog) 

v\       mg  +  Xe 

Applying  Stokes'  Law,   Eq.  (62),  to  the  sphere  whose  volume 
is  47rr3/3,  we  obtain 


(109) 


A  beautiful  application  of  this  method  has  been  made  by 
Millikan  (1909,  etc.).  He  has  found  the  most  probable  value 
for  e  to  be  4.69  X  10~10.  This  leads  to  the  number  of  molecules 
in  a  gram  molecule  N  =  6.18  X  1023  and  the  mass  of  the  hydro- 
gen atom  as  1.62  X  10~24  g. 

Chapman  (1916)  and  Rankine  (1920-1)  have  calculated  the 
diameters  of  the  atoms  of  the  monatomic  gases  from  determina- 
tions of  the  viscosity.  They  regard  the  atoms  as  hard  spheres 
having  the  well-defined  absolute  diameters  given  below. 

ATOMIC  DIAMETERS  OF  SOME  OP  THE  NOBLE  GASES  AFTER  RANKINE 


Gas 

Viscosity 

Crystal  measurement 

Neon... 

2.35  X  10-' 

61.30  X  10-' 

Argon  
Krypton  
Xenon  

2.87  X  10-' 
3.19  X  10-1 
3.51  X  10-1 

2.05  X  10-1 
2.35  X  10-» 
62.70  X  10-1 

These  values  agree  very  well  with  those  obtained  from  van 
der  Waal's  equation  but  they  are  somewhat  greater  than  the 
diameters  of  the  outer  electron  shells  of  the  atoms  as  obtained 
by  Bragg  from  his  crystal  measurements. 


CHAPTER  X 
SUPERFICIAL  FLUIDITY 

The  viscosity  of  a  liquid  may  change,  and  it  may  change 
in  a  quite  extraordinary  manner,  as  the  boundary  of  the  liquid 
is  approached.  This  must  of  necessity  result  wherever  the 
surface  tension  is  such  as  to  bring  about  a  change  in  concentra- 
tion at  the  boundary.  We  should  therefore  naturally  expect 
soap  and  saponin  solutions  to  show  this  phenomenon.  Experi- 
mentally this  field  of  study  has  not  been  much  explored  although, 
as  we  shall  attempt  to  show,  the  promise  of  reward  is  very  great 
and  the  need  of  such  study  in  industry  is  pressing.  However, 
Stables  and  Wilson  (1883)  have  proved  that  a  saponin  solution 
has  a  viscosity  at  the  surface  which  is  4,951  as  compared  with 
3.927  for  the  surface  of  pure  water.  The  viscosity  was  measured 
by  the  oscillations  of  a  circular  nickel-plated  brass  disk,  of  7.625 
cm  diameter  and  0.2  cm  thickness,  which  was  suspended  in  the 
liquid  by  means  of  a  wire  119.8  cm  long.  As  soon  as  the  solution 
was  allowed  to  rise  0.15  cm  above  the  disk  the  viscosity  fell  to  its 
normal  value. 

The  viscosity  found  by  Stables  and  Wilson  indicates  that 
the  surface  layer  of  a  supposedly  dilute  solution  may  nevertheless 
have  a  viscosity  which  is  over  a  thousand-fold  that  of  water 
at  20°C  (1,260  cp)  or  about  the  viscosity  of  castor  oil.  But  for 
very  small  stresses,  the  viscosity  may  be  still  higher,  for  it 
is  to  be  particularly  noted  that  in  a  saponin  solution  a  pendulum 
does  not  oscillate  isochronously.  Thus  in  one  experiment 
with  vibrations  of  large  amplitude,  Stables  and  Wilson  found  the 
time  of  vibration  to  be  10.52  seconds,  whereas  with  small  ampli- 
tudes the  time  of  vibration  was  9.73  sec.  This  would  indicate 
that  with  very  small  stresses  the  viscosity  might  be  found  to  be 
infinite,  which  would  mean  that  we  are  here  again  dealing  with 
plastic  flow. 

The  experiments  of  Stables  and  Wilson  need  confirmation  and 
254 


SUPERFICIAL  FLUIDITY  255 

extension  with  our  more  recent  knowledge  of  the  nature  of  flow 
in  mind,  but  whatever  the  surface  of  a  given  saponin  solution  may 
be,  we  may  profitably  distinguish  three  typical  cases:  (A)  where 
the  superficial  layer  is  a  true  solution  but  of  different  concen- 
tration from  the  interior  and  is  in  contact  with  it  own  vapor  or 
some  gas;  (B)  where  the  surface  is  made  up  of  a  layer  of  immisci- 
ble liquid,  which  may  be  so  thin  as  to  be  imperceptible  by  ordi- 
nary means;  (C)  where  the  surface  is  formed  either  by  a  continuous 
solid  or  by  solid  particles  in  more  or  less  intimate  contact  with 
each  other.  It  is  evident  that  in  the  last  two  cases  we  are  dealing 
not  with  the  superficial  fluidity  of  the  liquid  but  of  a  heterogene- 
ous mixture  of  liquid-liquid  or  solid-liquid  respectively. 

Soap  solutions  perhaps  afford  the  best  examples  of  the  first 
case  and  if  such  solutions  have  extraordinarily  high  superficial 
viscosity,  it  serves  to  explain  the  stability  of  the  soap  bubble. 
The  liquid  between  the  two  highly  viscous  surfaces  can  proceed 
downward  very  slowly  in  so  narrow  a  space. 

Oil  films  on  water  give  frequent  examples  of  the  second  sort, 
and  the  use  of  oil  "to  calm  troubled  waters"  is  a  practical  appli- 
cation of  superficial  viscosity  in  the  damping  of  vibrations.  The 
simple  harmonic  motion  of  the  wave  causes  the  particles  to  move 
in  vertical  circles,  so  that  an  oil  film  is  alternately  stretched  and 
compressed.  The  water  underneath  not  being  subjected  to  this 
same  tendency  is  pulled  along  by  the  oil  film  and  in  this  viscous 
flow  energy  is  of  course  dissipated.  A  method  for  the  measure- 
ment of  viscosity  by  Watson  (1902)  depends  upon  the  damping 
of  small  waves  in  a  free  surface,  and  apparently  this  method  is 
capable  of  being  used  to  measure  superficial  viscosity,  but  this 
appears  not  to  have  been  attempted. 

The  connection  of  superficial  fluidity  with  emulsions  must  be 
mentioned  at  this  point  although  we  cannot  stop  to  discuss  it. 
We  can  merely  refer  the  reader  to  the  fascinating  studies  of  Pla- 
teau, Quincke,  and  Lord  Rayleigh  upon  the  nature  of  contamina- 
ting films.  The  recent  paper  by  Irving  Langmuir  (1919)  on  the 
theory  of  flotation  is  very  suggestive. 

Many  of  the  examples  which  we  would  naturally  cite  as  exam- 
ples of  the  second  case  given  above  may  really  be  examples  of  the 
third  instead.  It  is  certain  that  in  most  emulsions  a  third  sub- 
stance is  necessary  to  stabilize  it  and  it  may  give  rigidity.  Scums 


256  FLUIDITY  AND  PLASTICITY 

are  apparently  examples  of  this  class.  Gurney  (1908)  in  investi- 
gating the  contamination  of  pure  water  surfaces  on  standing, 
says  "Water  surfaces  become  noticeably  rigid  in  a  few  hours  or 
days:  depending  on  the  previous  history  of  the  fluid.  Vigorous 
stirring  destroyed  the  rigidity  of  the  surface." 

To  prevent  possible  misunderstanding,  it  must  be  stated 
again  that  rigidity  in  foams  and  emulsions  arises  largely  from 
the  fact  that  during  shear  the  bubbles  of  a  foam  or  the 
globules  of  an  emulsion  are  distorted  and  may  be  disrupted, 
and  thus  work  is  done  against  the  forces  of  cohesion  opposing 
such  disruption. 

Superficial  viscosity  has  heretofore  been  considered  at  a  free 
surface  only.  Such  a  view  is  too  narrow  as  it  would  leave  the 
most  important  examples  out  of  consideration  and  from  the 
theoretical  aspect  the  extension  of  our  conception  of  superficial 
fluidity  involves  no  difficulty  whatever.  Having  made  this 
extension,  the  phenomenon  of  slipping  falls  into  the  third  case, 
but  the  fluidity  near  the  boundary  is  higher  than  that  of  the  main 
body  of  material.  Henry  Green  (1920)  has  studied  this  slippage 
under  the  microscope,  using  for  observation  paint  colored  with  a 
little  ultramarine,  which  may  be  subjected  to  shearing  stresses 
in  a  capillary  tube.  With  small  stresses  the  shear  takes  place 
exclusively  in  the  region  near  the  boundary,  but  when  the  stress 
becomes  greater  than  the  yield  value  of  the  paint,  the  shearing 
takes  place  throughout  the  material.  Green  reasons  that  it  is 
this  mixture  of  the  kinds  of  flow  which  causes-  the  shear  to  fail 
to  be  a  linear  function  of  the  shearing  stress,  particularly  when 
those  stresses  are  near  the  yield  shearing  stress.  In  the  above 
example,  the  layer  next  to  the  boundary  was  more  fluid  than  the 
mam  body  of  material,  but  more  often  the  opposite  is  the  case,  the 
fluid  near  the  boundary  is  less  fluid,  and  we  might  therefore  consider 
the  general  subject  of  adsorption  under  this  head.  And  we  would 
then  show  that  it  is  possible  to  make  a  fractional  separation  of 
fluids  by  simply  passing  them  through  capillary  tubes.  Such  a 
separation  of  a  mixture  into  its  components  by  means  of  capillary 
flow  has  actually  been  demonstrated,  as  in  the  case  of  petroleum 
forced  through  clay  by  Gilpin  and  his  co-workers  (1908).1  Since 
the  surface  area  of  a  capillary  varies  as  the  first  power  of  the 

1  Am.  Chem.  J.  40,  495  (1908);  44,  251  (1910);  50,  59  (1913). 


SUPERFICIAL  FLUIDITY  257 

radius  whereas  the  volume  of  flow  varies  as  the  square  of  the 
radius,  Eq.  (6),  we  may  expect  to  find  the  effects  of  superficial 
fluidity  shown  to  the  best  advantage  in  very  fine  tubes. 

There  are  a  variety  of  causes  which  may  cause  the  fluid  near 
the  boundary  to  have  a  different  fluidity.  The  most  important 
cause  results  from  the  selective  adhesion  of  the  components  of 
the  fluid  for  the  solid.  If  one  of  the  components  of  the  fluid  is 
more  strongly  attracted  than  another,  separation  becomes  possi- 
ble, and  the  magnitude  of  the  fluidity  of  the  mixture  as  measured 
will  theoretically  be  affected  The  adhesion  between  solid  and 
liquid  or  liquid  and  liquid  is  doubtless  just  as  specific  a  property 
as  is  the  better  known  cohesion  or  surface  tension  of  liquids  and 
we  are  coming  to  understand  the  nature  of  adhesion  better 
through  the  efforts  of  Langmuir  (1919)  and  Harkins  (1920).  We 
have  seen  that  it  is  possible  to  greatly  affect  both  the  friction 
and  the  mobility  of  plastic  substances  by  the  addition  of  small 
amounts  of  acid  or  alkali.  Just  what  happens  in  such  cases 
might  be  subject  to  dispute,  but  it  is  certain  that  small  amounts 
of  substances  adsorbed  on  to  the  surface  of  a  solid  may  entirely 
change  the  character  of  the  solid  which  is  in  contact  with  the 
liquid.  Thus  Henry  Green  (1920)  has  observed  that  the  addition 
of  small  amounts  of  gum  arabic  to  a  suspension  may  greatly 
decrease  the  yield  value  and  increase  the  mobility,  in  spite  of 
the  high  viscosity  of  gum  arabic  solutions.  This  is  interpreted 
as  being  due  to  the  decrease  in  adhesion  between  the  sus- 
pended particles.  The  well-known  work  of  Schroeder  (1903) 
upon  the  effects  of  electrolytes  on  the  viscosity  of  gelatine  and  of 
Handowsky  (1910)  upon  serum  albumin  should  also  be  referred 
to. 

We  have  already  proved  on  page  86  that  if  any  cause  results 
in  the  fluid  near  the  boundary  becoming  different  from  the  re- 
mainder of  the  liquid,  the  resulting  fluidity  will  be  changed. 
This  theorem  is  therefore  useful  in  explaining  superficial  fluidity. 
We  will  now  prove  that  the  components  of  a  mixture  under  these 
conditions  will  undergo  partial  separation.  The  conditions 
will  be  made  more  general  by  using  the  non-homogeneous  mixture 
considered  on  page  86.  Considering  the  mixture  as  made  up 
of  the  two  components  A  and  B,  arranged  in  alternate  plane 
layers,  the  total  quantity  of  A  flowing  in  a  unit  of  time,  regardless 


258  FLUIDITY  AND  PLASTICITY 

of  whether  it  is  derived  from  the  fluidity  of  A  or  B,  is  obtained 
from  the  terms  of  Eq.  (26)  containing  r\,  and  is 

2V1  n  I 

and  similarly  the  rate  of  flow  of  component  B  is 


There  will  be  separation  of  the  two  components  only  when  the 

thickness  of  the  different  layers  is  considerable  or  when  the 

passage  through  which  the  substances  are  forced  is  very  small, 

for  in  either  case  n  will  be  small.     If  n  =  « , 

Ui      a 

U2      b 

and  there  will  be  no  separation  at  all.  The  separation  may  be 
calculated  from  the  expression 

Ui      a  na<pi  +  (n  -  l)6g>2  ,     m 

U2      b'-(n  +  l)a?i  +nb<p2 

When  n  =  1,  the  component  A  will  flow  at  only  one-third  of 
the  rate  of  B,  even  though  the  two  components  have  the  same 
fluidity  and  are  present  in  equal  proportion;  and  even  if  the 
fluidity  of  B  is  zero,  it  will  flow  twice  as  rapidly  as  A,  under 
the  above  conditions.  It  follows  that  the  flow  of  B  is  greatly 
increased  by  making  the  fluidity  of  A  large,  this  being  the  layer 
in  contact  with  the  stationary  boundary. 

An  ingenious  application  of  the  principle  of  superficial  fluidity 
was  made  by  the  Southern  Pacific  Railroad,1  when  it  was  found 
that  the  pressure  required  to  pump  certain  heavy  oils  through 
long  pipe  lines  was  inconveniently  large.  The  problem  was 
to  get  the  maximum  flow  of  oil  for  a  given  expenditure  of  energy 
and  with  a  given  diameter  of  pipe.  By  using  a  rifled  pipe  and 
injecting  about  10  per  cent  of  water  along  with  the  oil,  the 
water  was  thrown  to  the  outside  of  the  pipe  by  the  centrifrugal 
action  caused  by  the  rifling,  producing  a  high  superficial  fluidity; 
and  thus,  by  a  seeming  paradox,  the  water  lubricated  the  oil  so 
that  the  delivery  became  from  8  to  10  times  what  it  would  have 
been  had  the  water  not  been  added. 

One  may  demonstrate  the  effect  of  superficial  fluidity  very 

1  Engineering  Record,  67,  676  (1908). 


SUPERFICIAL  FLUIDITY  259 

simply  by  comparing  the  times  required  by  gravity  to  empty  two 
pipettes  filled  with  a  heavy  oil,  each  of  the  pipettes  being  similar 
in  every  respect  except  that  one  is  moistened  with  water  previous 
to  filling  with  oil. 

In  an  experiment  by  the  author  at  25°C  and  a  pressure 
of  60  g  per  square  centimeter  a  given  volume  of  water  required 
33  sec.  and  the  same  volume  of  cottonseed  oil  required  1,640  sec. 
A  mixture  was  then  used  containing  one-third  oil  and  two-thirds 
water  by  volume.  Had  the  heavier  water  flowed  completely 
through  the  capillary  ahead  of  the  oil,  the  time  of  flow  should 
evidently  have  been  22  +  547  =  569  sec.;  yet  only  391  sec.  were 
actually  required  which  is  less  than  the  time  theoretically  required 
by  the  oil  alone.  The  difference  of  178  sec.  is  due  to  the  water 
forming  a  lubricating  film  for  the  oil  as  the  water  drained  out 
through  the  capillary. 

Rate  of  Absorption.  —  It  is  appropriate  here  to  show  how  the 
rate  of  absorption  of  a  fluid  into  a  porous  material  depends  upon 
the  fluidity  of  the  medium.  From  Poiseuille's  Law,  Eq.  (8),  it 

follows  that  the  rate  ^  at  which  a  liquid  enters  a  long  capillary 

tube  under  the  driving  force  P  will  be 

dl  =P#V. 

dt  ~     81 
If  the  capillary  is  very  small,  the  surface  tension  7  exerts  a  force 

2y 
P  which  must  be  added  to  the  external  pressure  and  this  force 

arising  from  the  surface  tension  may  be  so  great  that  the  external 
pressure  is  negligible  in  comparison,  in  which  case 

dl 

dt 

and  by  integration 


The  quantity  of  0.5<?7  is  called  the  coefficient  of  penetrance  of 
the  fluid  and  it  is  a  measure  of  the  tendency  of  a  liquid  to  pene- 
trate a  given  material  which  it  wets.  (Cf.  Washburn  Physical 
Chemistry,  2d  ed.,  p.  62.) 

The  distance  that  a  liquid  will  penetrate  a  given  porous  mate- 
rial due  to  capillary  action  alone  is  often  of  practical  importance. 


260  FLUIDITY  AND  PLASTICITY 

From  the  above  equation  we  see  that  this  distance  is  propor- 
tional to  the  square  root  of  the  fluidity,  the  surface  tension,  the 
radius  of  the  capillary  and  the  time. 

It  is  generally  assumed  that  the  material  of  the  pore  walls  is 
immaterial  so  long  as  the  walls  are  wet  by  the  liquid.  Adhesion 
between  solid  and  liquid  may  come  into  play  in  certain  cases 
making  such  an  assumption  fallacious,  as  already  pointed  out. 
Experiments  on  the  impregnation  of  fabrics,  belting,  wood  et 
cet.  with  oils,  gums,  paints  et  cet.  have  shown  that  thorough  dry- 
ing of  the  former  materials  has  an  extraordinary  effect  upon  the 
penetration  of  the  latter.  This  may  be  due  to  increasing  adhe- 
sion although  it  may  be  explained  in  some  other  way. 


CHAPTER  XI 
LUBRICATION 

When  a  solid  substance  is  subjected  to  a  shearing  stress, 
it  undergoes  plastic  flow  if  the  stress  is  greater  than  the  yield 
value  of  the  material.  In  this  process  of  shear,  lateral  stresses 
arise  and  if  the  material  is  not  supported  laterally  by  sufficient 
pressure,  rupture  of  the  material  will  finally  result.  These 
surfaces  formed  by  rupture  slide  over  each  other  according  to  the 
laws  of  solid  friction  stated  by  Coulomb.  The  surfaces  are 
separated  for  the  most  part  by  a  layer  of  fluid  which  may  be  air, 
water,  oil,  a  layer  of  oxide,  etc.  So  two  surfaces  formed  by  a 
rupture  as,  for  example,  two  broken  pieces  of  porcelain  do  not 
adhere  together  firmly  even  when  they  seem  to  fit  together  very 
nicely.  So  also  the  resistance  to  movement  between  ordinary 
smooth  surfaces  is  far  less  than  the  resistance  to  plastic  flow. 

If,  however,  sufficient  force  is  brought  to  bear  between  two 
sliding  surfaces  of  similar  material,  there  will  occur,  far  below 
the  melting  point  of  the  substance,  a  welding  together  of  the 
surfaces  into  a  more  or  less  compact  whole,  unless  there  is 
present  some  substance  which  prevents  such  welding.  Two 
surfaces  of  glass  ordinarily  touch  each  other  at  very  few  points 
and  they  do  not  adhere  strongly,  but  when  the  two  surfaces 
are  ground  to  an  optical  surface  and  cleaned,  it  is  difficult  to 
separate  the  two  surfaces  without  tearing  them,  after  they  have 
been  brought  together.  A  motor  bearing  which  has  been  care- 
fully fitted  by  "lapping  in"  may  be  ruined  completely  by  a 
slight  turn  with  the  hand  after  the  surfaces  have  been  cleaned 
and  again  brought  together.  Powdered  metals  adhere  strongly 
when  subjected  to  heavy  pressures,  even  at  temperatures  con- 
siderably below  the  melting  point.  The  Johannsen  blocks  used 
in  gage  testing  are  made  of  hardened  steel  with  surfaces  which  are 
exceptionally  true.  When  these  blocks  are  placed  one  on  top  of 
the  other,  the  adhesion  between  them  is  so  great  that  a  pile  of 

261 


262  FLUIDITY  AND  PLASTICITY 

them  several  inches  high  can  be  raised  by  lifting  the  topmost 
one.  In  imperfect  lubrication  we  first  have  excessive  wear,  then 
scoring  of  the  bearings  and  finally  seizure  with  a  more  or  less 
complete  welding  together  of  the  surfaces.  Thus  there  is  a  con- 
siderable mass  of  evidence  to  prove  that  whenever  two  clean 
surfaces  come  together  they  adhere  and  thus  the  conditions  for 
plastic  flow  may  be  reestablished.  The  problem  of  lubrication 
is  therefore  to  substitute  as  far  as  possible  fluid  friction  for  the 
enormously  higher  resistance  to  shear  in  plastic  flow. 

According  to  the  above  view,  "solid  friction,"  as  ordinarily 
observed,  is  intermediate  between  true  plastic  flow  and  true 
viscous  flow.  Under  favorable  conditions  it  approaches  closely 
to  simple  viscous  flow,  whereas  under  very  unfavorable  conditions 
it  may  approach  the  conditions  for  plastic  flow.  It  is  clear 
therefore  that  the  coefficient  of  solid  friction  may  vary  within  the 
widest  limits  depending  upon  the  condition  of  the  bearing 
surfaces,  the  temperature,  speed,  and  character  of  the  lubricant. 

Thus  at  the  outset  we  may  state  that  it  is  impossible  to 
specify  the  lubricant  that  will  be  most  suitable  for  a  given 
machine,  provided  that  that  machine  works  at  variable  speeds, 
temperatures  and  loads,  and  where  the  bearings  are  continually 
subject  to  wear  due  to  defective  lubrication.  On  the  other 
hand,  if  bearings  are  perfectly  lubricated  and  run  under  constant 
conditions,  there  is  practically  no  wear,  so  that  the  problem  to 
find  the  most  suitable  lubricant  has  a  definite  solution.  With 
the  steady  advance  of  industrial  development,  the  theory  of 
lubrication  takes  on  increasing  interest. 

The  laws  of  solid  friction  may  be  stated  as  follows:  (1)  When 
two  unlubricated  smooth  surfaces  slide  over  each  other,  the  fric- 
tional  resistance  P  varies  directly  as  the  load  W  or 

P  =  f  W  (111) 

and  the  coefficient  of  friction  f  is  defined  as  the  ratio  between  the 
friction  and  the  load. 

2.  The  force  P0  required  to  maintain  an  indefinitely  small  rate 
of  shear,  the  so-called  static  friction,  is  greater  than  when  the 
rate  of  shear  is  appreciable.     The  dynamic  friction  is  independent 
of  the  velocity. 

3.  The  friction  is  independent  of  the  area  of  the  surfaces  in 


LUBRICATION  263 

apparent  contact,  within  wide  limits.  The  surfaces  must,  how- 
ever, be  large  enough  so  that  the  surfaces  remain  intact. 

Since  it  is  impracticable  to  obtain  a  pair  of  smooth  and  entirely 
unlubricated  surfaces,  it  is  needless  to  say  that  these  laws  are  very 
inexact.  As  already  intimated,  well-fitting  and  clean  surfaces  of 
similar  material  would  probably  seize  and  follow  the  laws  of  plastic 
flow,  which  are  very  different  from  the  laws  given  above.  They 
have,  however,  both  historic  interest  and  practical  usefulness. 

Just  as  the  laws  of  solid  friction  are  superficially  unrelated 
to  the  laws  of  plastic  flow,  so  these  laws  are  also  in  sharp  contrast 
to  the  laws  of  viscous  flow  which  apply  to  well-lubricated  surfaces. 
With  well-lubricated  surfaces  we  have  the  relation 


where  S  is  the  area  of  surface  in  contact,  dv  is  the  velocity  and  dr 
is  the  thickness  of  the  oil  film.     According  to  this  relation: 

1.  The  frictional  resistance  P  is  independent  of  the  load. 

2.  The  friction  is  directly  proportional  to  the  velocity  and  is 
therefore  zero  when  the  velocity  is  zero. 

3.  The  friction  is  also  directly  proportional  to  the  area  of  sur- 
faces in  contact. 

In  view  of  the  absolute  antithesis  between  these  two  sets  of 
laws,  it  is  not  surprising  that  the  results  of  the  study  of  friction 
as  recorded  in  the  literature  are  often  contradictory.  We  may, 
however,  state  broadly  that  slow-moving,  poorly  lubricated  sur- 
faces follow  approximately  the  laws  of  solid  friction,  whereas 
rapid-moving  and  hence  necessarily  well-lubricated  machinery, 
such  as  electric  dynamos  and  motors,  follows  the  laws  of  fluid 
friction.  Most  bearings  are  imperfectly  lubricated  and  follow 
neither  set  of  laws  exactly. 

Petroff  (1887)  seems  first  to  have  applied  the  laws  of  fluid 
friction  to  lubricated  bearings  testing  out  his  views  by  experiment. 

Most  important  in  its  relation  to  the  development  of  the  theory 
of  lubrication  is  the  experimental  work  of  Beauchamp  Tower 
(1883-4),  undertaken  at  the  instance  of  the  Institution  of 
Mechanical  Engineers.  His  experiments  were  conducted  with 
extreme  care  and  under  varied  and  well-chosen  circumstances. 
His  results,  as  obtained  under  ordinary  conditions  of  lubri- 


264  FLUIDITY  AND  PLASTICITY 

cation,  "so  far  agree  with  the  results  of  previous  investigators  as 
to  show  the  want  of  any  regularity."  He  perceived  that  this 
difficulty  was  due  to  irregularity  in  the  supply  of  lubricant,  so 
he  conducted  experiments  in  an  oil  bath.  Not  only  was  he  thus 
able  to  obtain  a  high  degree  of  regularity  but  he  proved  that  the 
journal  and  bearing  are  completely  and  continuously  separated 
by  a  film  of  oil.  This  film  is  maintained  by  the  motion  of  the 
journal  against  a  hydrostatic  pressure  in  the  oil,  which  at  the  crown 
of  the  bearing  was  shown  by  actual  measurement  to  be  625  Ib. 
per  square  inch  greater  than  the  pressure  in  the  oil  bath. 

Tower  demonstrated  that  even  with  an  oily  pad  in  contact 
with  the  journal,  the  results  were  regular  although  the  results 
were  different  from  those  with  the  oil  bath.  Of  lubrication  less 
than  that  afforded  by  the  oil  pad  he  says :  "  The  results,  generally 
speaking,  were  so  uncertain  and  irregular  that  they  may  be  sum- 
med up  in  a  few  words.  The  friction  depends  on  the  quantity 
and  uniform  distribution  of  the  oil,  and  may  be  anything  between 
the  oil  bath  results  and  seizing,  according  to  the  perfection  or 
imperfection  of  the  lubrication." 

These  experiments  of  Tower  are  indeed  a  landmark  in  the 
development  of  the  theory  of  lubrication  for  they  stimulated 
various  investigators  such  as  Osborne  Reynolds,  Stokes,  and  Lord 
Rayleigh  to  apply  the  fundamental  hydrodynamical  equations 
to  the  results  obtained.  And  the  labors  of  Reynolds,  continued 
by  Sommerfeld  (1904)  and  Michell  (1905),  have  in  fact  enabled 
us  to  reach  a  complete  solution  of  the  problem  of  lubrication  in 
certain  very  special  cases.  The  mathematical  integrations  have 
generally  proved  very  difficult. 

REYNOLDS'  THEORY  OF  LUBRICATION 

The  model  of  viscous  flow  which  we  have  considered,  page  5, 
does  not  give  rise  to  any  pressure  at  right  angles  to  the  direction 
of  flow,  hence  it  is  unable  to  sustain  a  load  permanently  and  will 
not  serve  for  practical  lubrication. 

Case  I.  Parallel  Surfaces  A  pproaching  with  Tangential  Motion. 
Let  AB  in  Fig.  83  represent  the  section  of  a  surface  which  is 
moving  with  the  uniform  velocity  U  in  respect  to  the  bearing 
block  CD,  each  being  of  indefinite  length  in  the  direction  perpen- 


LUBRICATION  265 

dicular  to  the  paper.  As  soon  as  a  load  is  placed  on  the  bearing 
block,  the  liquid  begins  to  be  squeezed  out  from  between  the 
surfaces.  If  this  space  is  divided  originally  into  the  equal  areas 
indicated  by  the  dotted  lines,  these  lines,  moving  with  the  fluid, 
will  after  a  time  occupy  the  positions  of  the  curved  lines;  and 
the  distances  moved  by  the  particles  are  shown  by  the  distances 
between  the  corresponding  points  on  the  two  sets  of  curves,  as 
QP  for  the  point  P,  and  the  slopes  of  the  curves  indicate  the  direc- 
tions of  the  forces  in  the  fluid  just  as  if  the  lines  were  stretched 


FIG.  83. — The  simplest  case  of  lubrication.     Two  parallel,  plane  surfaces. 

elastic  threads.  The  pressures  exerted  upon  different  points 
along  CD  are  shown  in  the  curve  of  pressures  CFD,  the  pressures 
being  proportional  to  the  vertical  height  above  the  line  CED. 
At  the  center  of  the  block  the  pressure  is  a  maximum  and  the 
liquid  is  squeezed  out  to  the  right  and  left  of  this  section.  For 
this  section  alone,  there  is  a  uniform  variation  of  velocity  from 
A B  to  CD,  such  as  would  be  true  of  all  sections,  if  the  surfaces 
AB  and  CD  were  not  approaching. 

Case  II.  Surfaces  Inclined — Tangential  Movement  Only. — 
If  now  the  bearing  block  is  tilted,  we  have  fulfilled  the  necessary 
condition  for  continuous  lubrication,  for  the  bearing  is  able 
to  sustain  a  load  without  the  surfaces  approaching  each  other. 

Were  we  to  assume  that  in  this  case  the  velocity  varies  uni- 
formly from  U  at  AB  to  zero  at  CD,  the  quantity  of  fluid  passing 
any  cross-section  MN  would  be  proportional  to  MN  X  U/2, 
or  simply  to  MN.  But  since  the  quantity  of  fluid  passing  every 
cross-section  must  be  the  same,  there  must  be  an  outflow  to 
the  right  and  left  of  the  cross-section  M'N',  at  which  the  pressure 
is  a  maximum,  so  the  flow  at  any  section  MN  is 

(MN  -  M'N')U/2 
At  the  cross-section  MN,  the  velocity  varies  uniformly  from 


266 


FLUIDITY  AND  PLASTICITY 


AB  to  CD,  but  the  point  of  maximum  pressure  M  is  not  at  the 
center  of  the  block  nor  is  it  necessarily  the  point  of  application 
of  the  resultant  pressure  exerted  on  the  block. 

If  the  bearing  is  free  to  move,  it  will  move  either  up  or  down 
until  the  pressure  is  just  equal  to  the  load.  As  the  load  is 
increased,  the  surfaces  approach  each  other,  which  increases 
the  friction  and  thereby  the  pressure  so  that  equilibrium  is 
restored.  But  the  point  of  application  of  the  resultant  pressure 
changes  with  the  load  provided  that  the  inclination  of  CD  remains 
the  same. 

Case  III.  Revolving  Cylindrical  Surface — Bearing  Surface 
Flat. — The  curves  of  motion  are  represented  in  Fig.  84.  To  the 


A  B' 

FIG.   84. — Simple  continuous  lubrication. 

right  of  GH  which  is  the  point  of  nearest  approach  of  the  sur- 
faces, the  curves  are  similar  to  those  in  Case  II.  At  the  left  of 
GH,  the  curves  are  quite  the  reverse  of  those  on  the  right,  being 
convex  toward  a  section  M2Nz  on  either  side,  just  as  they  are 
concave  to  a  section  MiN\  on  the  right.  The  reason  for  this  is 
that  with  a  uniformly  varying  velocity  more  fluid  would  be 
brought  in  at  the  right  of  MiNi  than  would  pass  the  section  GH, 
hence  the  fluid  must  flow  outward  from  MiNi,  where  the  pressure 
is  a  maximum  in  both  directions.  So  at  the  left  of  GH  more 
fluid  would  be  carried  away  than  arrives  through  GH,  hence  an 
inflow  is  necessary  to  the  right  and  left  of  the  section  of  minimum 
pressure  M^N^.  The  fluid  pressure  acts  to  separate  the  surfaces 
at  the  right  and  to  draw  them  together  at  the  left  hence  there  is  a 
couple  of  forces  resulting. 

If  the  bearing  is  cut  away  at  the  left  of  GH,  the  negative  pres- 
sures may  be  eliminated. 


LUBRICATION 


267 


If  the  oil  supply  is  limited,  the  oil  may  not  wet  the  entire 
bearing  but  form  an  oil  pad  in  the  region  of  GH,  the  pressures  of 
course  reaching  a  zero  value  at  the  points  where  the  oil  surface 
meets  the  bearing  surface.  If  d  is  the  thickness  of  the  oil  film 
outside  of  the  pad,  the  quantity  brought  up  to  the  pad  per 
second  will  be  Ud,  and  the  quantity  which  passes  the  section 
MiNi  where  the  velocity  varies  uniformly  is  MiNiU/2,  and  since 
there  is  no  accumulation  of  oil,  these  two  values  must  be  equal 
and 

MlNl  =  2d 
also  MZNZ  =  2d 


Case  IV.  Revolving  Cylindrical  Surface  —  Bearing  also  Cylin- 
drical. —  In  a  very  common  example  of  lubrication  we  have  a 
cylindrical  journal  partly  or  wholly  surrounded  with  the  bearing 
or  "  brass  "  CD  in  Fig.  85.  The  oil  is  drawn  up  into  the  space  BD 


FIQ.  85. — The  lubricated  journal  and  bearing. 

and  creates  a  pressure  which  is  a  maximum  at  Q.  The  point  of 
nearest  approach  between  journal  and  bearing  is  not  at  the 
middle  of  the  bearing  0  but  at  a  point  some  40°  further  on 
at  G  toward  the  so-called  " off-side"  of  the  bearing.  This  is 
the  opposite  to  what  happens  in  the  unlubricated  bearing,  for 


268  FLUIDITY  AND  PLASTICITY 

the  point  of  nearest  approach  is  then  on  the  "on-side."  Only 
when  the  bearing  is  unloaded  does  the  point  of  nearest  approach 
coincide  with  the  middle  of  the  brass,  O.  As  the  load  increases 
the  point  G  moves  from  0  up  to  a  certain  maximum  value  after 
which  it  recedes  toward  0,  resulting  finally  in  a  discontinuity  in 
the  oil  just  as  in  the  case  of  a  limited  supply  of  oil. 

We  have  considered  only  bearings  of  unlimited  length,  whereas 
in  practical  bearings  the  lubricant  is  squeezed  out  at  the  sides, 
as  well  as  at  the  ends.  Michell  (1905)  has  made  a  study  of  the 
changes  of  pressure  in  the  oil  film  of  bearings  of  various  shapes. 
Generally  speaking  the  integrations  necessary  to  define  the  exact 
relations  between  load,  speed  and  the  friction  have  not  been 
effected. 

The  theory  of  lubrication  is  not  inconsistent  with  the  experience 
that  the  friction  in  limited  lubrication  is  proportional  to  the 
load  and  independent  of  the  velocity.  Increase  of  load  will 
result  in  a  diminution  of  the  distance  between  the  bearing 
surfaces,  a  lengthening  of  the  oil-pad,  and  therefore  an  increase 
in  the  resistance.  Increasing  the  velocity  increases  also  the 
resistance,  but  it  also  increases  the  pressure  and  therefore  the 
distance  between  the  surfaces,  provided  that  the  load  is  kept 
constant,  and  this  produces  a  decrease  in  the  resistance. 

For  further  details  of  the  development  of  this  very  important 
subject  the  reader  is  referred  to  the  original  papers  of  Petroff, 
Tower,  Reynolds,  Sommerfeld,  Michell,  Lasche  to  name  but  a  few. 

LUBRICATION  AND  ADHESION 

In  the  early  use  of  lubrication,  fixed  oils  and  greases  were 
depended  upon  almost  exclusively.  The  fixed  oils,  that  is 
the  non-volatile  oils  of  animal  or  vegetable  origin,  are  expensive, 
they  may  become  gummy  and  rancid,  which  interferes  with 
proper  lubrication  and  the  acids  developed  may  corrode  the 
machines.  These  oils  moreover  often  partially  solidify  when  only 
slightly  cooled.  The  range  of  viscosities  obtainable  is  also 
restricted  by  the  small  number  of  oils  available  in  any  quantity. 
With  the  advent  of  mineral  oils,  these  troubles  were  all  overcome, 
so  the  battle  which  was  waged  between  the  mineral  and  the 
fixed  or  fatty  oils  was  short  and  apparently  decisive.  The 


LUBRICATION  269 

purveyors  of  the  fatty  oils  claimed  that  their  oils  possessed 
greater  "oiliness,"  "body"  or  " lubricating  value,"  but  since 
these  claimants  seem  never  to  have  proved  their  case  by  the 
actual  measurement  of  "oiliness"  and  since  modern  indus- 
trialism requires  vastly  more  oil  for  lubrication  than  could  pos- 
sibly be  met  by  the  available  supplies  of  fatty  oils,  the  conception 
of  the  property  of  oiliness  has  gradually  become  a  sort  of  will  o' 
the  wisp  vaguely  referred  to  in  treatises  on  lubrication,  and  effec- 
tively used  by  energetic  salesmen  in  convincing  a  prospective 
buyer  of  the  superiority  of  a  given  brand  of  oil  over  all  others. 
The  theory  predicted  that  so  long  as  the  viscosity  was  sufficient  to 
produce  the  necessary  pressure  required  to  carry  the  load,  it 
was  of  no  moment  what  the  chemical  nature  of.  the  lubricant 
might  be,  provided  only  that  the  quantity  of  lubricant  was  ample. 
The  practice  has  therefore  been  to  use  an  oil  which  is  much  more 
viscous  than  is  really  necessary  and  to  accept  a  loss  in  power 
in  order  to  insure  against  any  discontinuity  in  the  oil  film. 

There  are,  to  be  sure,  many  instances  which  might  be  cited 
where  an  experienced  engineer  has  cooled  a  hot  bearing  by 
substituting  a  fixed  oil  with  which  he  was  familiar  for  the  mineral 
oil  in  use.  However,  in  comparing  two  oils  used  for  practical 
lubrication,  there  are  so  many  factors  which  may  affect  the 
comparison  such  as  the  quantity  of  oil,  the  speed,  load,  tem- 
perature of  the  oil  film,  the  condition  of  the  bearing  surfaces, 
that  instances  which  might  be  cited  are  easily  discredited  by  the 
skeptical.  Nevertheless,  there  is  a  growing  demand  for  lubri- 
cants which  will  be  less  wasteful  of  power  and  which  will  at  the 
same  time  give  the  maximum  assurance  that  "the  bearings  will  not 
be  injured  in  use.  With  the  aeroplane  in  particular,  it  is  neces- 
sary to  keep  the  motor  going  at  all  hazards  during  most  of  the 
period  of  flight,  and  an  overheated  bearing  may  easily  cause  the 
complete  wreckage  of  the  machine  in  mid-air,  so  the  selection 
of  the  best  lubricant  for  severe  conditions  and  the  question  of 
"oiliness"  becomes  now  vitally  important.  Perhaps  the  clearest 
evidence  on  this  point  is  obtained  from  cutting  lubricants. 

Cutting  Lubricants. — It  is  the  well-nigh  universal  testimony 
of  mechanicians  that  in  certain  cutting  operations,  fixed  oils  are 
absolutely  necessary  and  that  mineral  oils  will  not  serve  as  a 
satisfactory  substitute.  Voluminous  correspondence  with  large 


270  FLUIDITY  AND  PLASTICITY 

shops  aH  over  this  country,  with  concurring  evidence  from  Great 
Britain,  establishes  the  fact  that  fixed  oils,  preferably  lard  oil, 
are  superior  to  all  others.  This  is  particularly  true  in  operations 
such  as  "parting  off"  soft  steel,  in  threading  wrought  iron  or 
steel,  in  drilling  deep  holes  in  steel  as  in  the  manufacture  of  gun 
barrels.  The  tool  keeps  its  edge  longer,  the  machine  runs  more 
smoothly,  there  is  less  heating,  a  much  greater  speed  may  be 
attained,  the  chip  is  less  serrated  and  therefore  longer,  the  cut 
surface  is  smoother  and  much  closer  dimensions  may  be  obtained, 
when  using  lard  oil  or  its  equivalent. 

On  the  other  hand,  there  are  certain  operations  such  as  planing 
and  reaming  where  a  lubricant  is  not  required.  In  others  such 
as  sawing  metals  a  liquid  may  be  used  merely  to  cool  the  work. 

No  lubricant  is  ordinarily  used  in  cutting  cast  iron,  brass  or 
aluminum.  Wrought  iron  and  "draggy"  metals  require  a 
lubricant. 

Between  the  two  extremes  of  those  operations  and  materials 
which  absolutely  require  a  fixed  oil  and  those  which  require  no 
liquid  at  all,  there  are  a  great  number  of  classes  of  work  in  which 
mineral  oils  are  satisfactory  but  where  aqueous  soap  solutions 
or  oil-emulsions  are  widely  used  and  found  to  be  highly  satisfac- 
tory. In  these  cases  the  oil  or  water  serves  to  reduce  the  heating 
of  the  work  and  the  tool,  and  the  soap  or  soda  prevents  the  rust- 
ing of  the  machine.  Fixed  oils  are  often  a  needless  extravagance 
or  positively  disadvantageous. 

Where  lard  oil  is  required  it  is  not  primarily  to  conduct  away 
the  heat,  for  the  operation  may  be  a  light  surfacing  operation 
where  the  heat  developed  is  slight  as  in  the  cutting  of  fine  micro- 
meter screws.  Its  superiority  does  not  depend  on  its  peculiar 
viscosity  because  a  mineral  oil  possessing  the  same  viscosity  in 
no  way  shares  its  superiority. 

It  is  true  that  mineral  oils  increase  in  fluidity,  when  heated, 
more  rapidly  than  fatty  oils,  but  castor  oil  is  exceptional  in  this 
respect  resembling  the  mineral  oils  and  yet  it  appears  to  be  a  very 
useful  cutting  oil  and  lubricant. 

It  has  also  been  suggested  that  pressure  might  decrease  the 
fluidity  of  the  mineral  oils  less  rapidly  than  that  of  the  fixed  oils, 
but  this  explanation  appears  to  be  not  even  qualitatively  correct 
(cf.  page  89,  Report  of  the  Lubricants  and  Lubrication  Inquiry 


LUBRICATION  271 

Committee.     Department  of  Science  and  Industrial  Research. 
(London)). 

The  surface  tensions  of  mineral  and  of  fixed  oils  are  not  materially 
different.  These  oils  are  however  very  different  in  one  important 
respect  viz.,  that  the  fixed  oils  all  have  an  active  chemical  group 
which  gives  them  a  strong  adhesion  for  metals,  so  that  such  an 
oil  is  not  readily  squeezed  out  from  between  two  metallic  sur- 
faces (cf.  Langmuir  (1919),  Harkins  (1920),  and  Bingham  (1921)). 
Lord  Rayleigh  (1918)  has  shown  that  a  layer  of  lubricant  of 


FIG.  86. — Illustration  of  the  necessity  for  high  adhesion  in  an  oil  which  is  to 
have  the  best  lubricating  quality. 

monomolecular  thickness  possesses  truly  remarkable  properties 
in  reducing  the  friction  between  solid  bodies  of  similar  material, 
the  contamination  probably  serving  to  prevent  the  welding 
together  of  the  surfaces.  According  to  Langmuir  (1920)  such 
a  film  formed  from  paraffin  oil  can  be  readily  removed  by  a  gentle 
stream  of  running  water  from  platinum,  glass,  etc.,  but  a  film 
formed  by  oleic  acid  cannot  be  thus  removed. 

To  get  a  clearer  idea  of  the  action  of  a  cutting  lubricant  we  will 
follow  the  operation  of  an  Armstrong  parting  tool  in  cutting  off 
disks  from  a  rod  of  soft  steel  1%  in.  in  diameter,  using  a  lathe  with 
a  constant  speed  and  feed  and  as  lubricants  a  definite  amount  of 
lard  oil  or  of  mineral  oil  of  the  same  viscosity.  Some  30  disks 
were  made  with  lard  oil  and  at  the  end  there  was  no  evidence 


272 


FLUIDITY  AND  PLASTICITY 


that  the  operation  was  not  as  satisfactory  as  at  the  beginning. 
The  disk  shown  in  the  left  of  Fig.  86  was  perfectly  smooth,  and 
there  was  little  evidence  of  heating  and  on  inspection  the  tool 
was  found  to  be  not  even  slightly  dulled.  The  chips  shown  below 
the  disk  were  only  slightly  serrated. 

On  substituting  the  mineral  oil  heating  began  at  once,  the 
surface  of  the  disk  shown  at  the  right  of  the  figure  was  very 
rough,  the  chips  were  deeply  serrated,  and  the  tool  so  dulled  that 
it  failed  completely  on  cutting  the  fourth  disk.  On  examining 
the  cut  in  the  bar  at  the  time  of  failure,  shown  in  the  middle  of  the 


FIG.  87. — Illustration  of  the  forming  of  a  chip  in  the  cutting  of  metals  and  of 
the  function  of  the  lubricant. 

figure,  one  can  plainly  see  two  beads  of  metal  flowing  ahead  of 
the  tool  and  gouging  into  the  bottom  of  the  cut.  A  burr  is 
being  thrown  up  at  the  left. 

The  operation  of  a  tool  in  cutting  is  illustrated  diagrammati- 
cally  in  Fig.  87.  The  metal  6  is  being  cut  away  by  the  tool  c,  a 
chip  /  being  formed  which  bears  down  heavily  upon  the  tool  at 
a  point  d  some  distance  back  from  the  point.  That  this  is  the 
actual  case  is  proved  by  many  facts.  For  example,  a  tool  in  use 
is  often  gouged  out  by  the  shaving  at  some  distance  back  from 
the  point,  and  there  is  sometimes  found  a  "bead"  of  metal 


LUBRICATION  273 

welded  to  the  tool  at  this  point.  The  tool  therefore  pries  the 
chip  away  rather  than  cuts  it,  and  the  point  of  the  tool  merely 
clears  up  the  surface,  so  long  as  the  tool  is  well  lubricated. 

The  surface  of  the  chip  is  serrated  and  of  about  twice  the 
thickness  of  the  cut.  We  have  here  evidently  a  case  of  plastic 
flow.  The  explanation  of  the  serrations  and  the  thickening 
is  probably  as  follows: — As  the  tool  moves  into  the  metal,  the 
strain  gradually  increases  and  a  certain  accommodation  takes  place 
due  to  the  elasticity  of  the  metal  and  the  machine.  When  the 
shearing  stress  reaches  the  yield  point,  the  metal  flows,  and  the 
more  rapidly  as  the  temperature  rises  rapidly  in  the  region  of 
flow.  In  this  process  the  pressure  on  the  tool  is  relieved,  the 
stress  falls  again  below  the  yield  point,  and  the  process  is  repeated. 
If  the  machine  is  very  sturdy  with  very  little  play,  the  cutting 
will  be  steadier,  but  here  comes  the  advantage  in  the  use  of  a 
good  lubricant,  that  it  is  drawn  into  the  space  ra,  contaminates 
the  under  side  of  the  freshly  formed  surface  of  the  chip  and  there- 
fore substitutes  viscous  flow  for  the  energy-consuming  plastic 
flow  to  a  greater  or  less  degree  depending  upon  the  efficiency  of 
the  lubricant  (cf.  Taylor,  "The  Art  of  Cutting  Metals"). 

If  the  lubrication  is  not  effective,  the  pressure  on  the  tool  must 
be  relieved  to  a  greater  extent  by  means  of  plastic  flow  of  the 
material.  The  result  is  greater  fluctuations  in  pressure,  the  metal 
flowing  outward  during  the  period  of  flow,  producing  serrations 
of  increased  height,  and  possibly  flowing  downward  into  the 
space  m.  It  is  this  metal,  flowing  inward  toward  the  work  and 
the  point  of  the  tool  which  creates  the  most  serious  condition, 
for  it  tends  to  break  off  the  edge  of  the  tool  and  to  gouge  into  the 
face  of  the  work. 

With  brittle  substances  such  as  cast  iron,  it  is  readily  per- 
ceived why  a  lubricant  is  not  necessary.  The  chip  breaks 
as  it  is  pried  off  and  there  is  comparatively  little  if  any  plastic 
flow.  In  cutting  very  hard  and  brittle  materials  such  as  glass 
and  some  varieties  of  steel,  a  lubricant  as  such  is  not  needed,  but 
something  which  perhaps  has  just  the  opposite  property  of 
causing  the  tool  to  adhere  to  the  material,  i.e.,  will  cause  the 
tool  to  "take  hold"  or  "bite."  Turpentine  is  used  for  this 
purpose  on  steel  and  turpentine  with  or  without  camphor  is 
used  on  glass.  It  is  difficult  to  see  how  these  substances  act 


274  FLUIDITY  AND  PLASTICITY 

unless  they  serve  to  remove  the  contaminating  film  of  greaso 
which  is  already  present. 

These  results  lead  one  to  the  observation  that  in  difficult  cases 
of  lubrication,  where  seizure  is  always  possible  and  is  almost 
certain  to  be  very  disastrous,  the  use  of  pure  mineral  oil  may  not 
be  the  best  practice.  On  the  other  hand,  there  is  not  enough  of 
the  fixed  oils  to  supply  the  imperative  demands  of  mankind  for 
edible  fats,  soaps,  leather  dressing,  et  cet.  Fortunately  however 
it  is  likely  that  all  of  the  benefit  of  the  use  of  lard  oil  as  a  lubricant 
can  be  obtained  very  cheaply  by  adding  to  mineral  oils  small 
amounts  of  certain  substances  possessing  high  adhesion,  par- 
ticularly substances  with  unsaturated  groups  in  their  molecules, 
such  as  are  found  in  oleic  acid,  turpentine,  pine  oil  et  cet.  Some 
of  these  substances  are  already  being  used  on  a  somewhat 
extensive  scale  in  successful  substitutes  for  cutting  oils.  The 
use  of  these  substitutes  opens  up  a  field  for  research  which 
is  most  fascinating  and  in  view  of  the  approaching  exhaustion 
of  our  supplies  of  petroleum,  the  study  is  so  practical  that  it 
cannot  long  be  postponed.  Of  its  importance  we  can  do  no 
better  than  quote  from  an  editorial  in  the  Chemical  Trade 
Journal  for  December  1920:  "Before  the  war  the  annual  expendi- 
ture on  lubricants  in  England  was  £6,000,000  and  it  is  estimated 
that  an  annual  saving  of  one  to  two  millions  could  be  effected 
if  a  systematic  investigation  were  undertaken  and  the  results 
made  freely  available  to  the  public.  Furthermore  the  loss 
caused  by  improper  lubrication,  would  represent  a  very  large 
addition  to  the  figure  given  above." 

Asphalt-base  Versus  Paraffin-base  Oils. — With  lubricants  in 
use  made  from  crude  oils  from  different  fields,  the  question  has 
arisen  whether  the  paraffin-base  or  the  asphalt-base  oil  is  supe- 
rior, but  there  is  a  notable  lack  of  convincing  evidence  in  favor 
of  either.  We  offer  the  following  evidence  to  prove  that  the 
differences  between  them  may  be  very  considerable,  and  that 
the  chemical  composition  as  determined  by  the  source  of  the 
oil  is  not  a  matter  of  indifference  to  the  consumer;  this  is  par- 
ticularly true  in  aeroplane  lubrication  where  the  results  of  faulty 
lubrication  are  so  very  disastrous.1 

1  The  walls  of  the  aeroplane  motor,  the  crankshaft  et  cet.  are  made  so 
light  that  the  seizure  of  a  single  bearing  will  result  in  the  wrecking  of  the 


LUBRICATION 


275 


Hexane  (CeHn)  represents  a  typical  paraffin-base  hydro- 
carbon, diallyl  (CeHio)  may  be  taken  to  represent  an  unsaturated 
non-cyclic  hydrocarbon,  whereas  benzene  (C6H6)  and  hexa- 
methylene  (CeH^)  represent  types  of  cyclic  hydrocarbons.  All 
of  these  compounds  have  the  same  number  of  carbon  atoms,  but 


50 


0°     10°     ^0U    30°     40°    50°    60°     70°    80°     90°     100°  110° 
Temperature,  Centigrade 

FIG.  88. — A    comparison    of    the  fluidity-temperature  curves  of  hydrocarbons 
of  different  homologous  series. 

whereas  their  fluidity-temperature  curves  are  nearly  parallel, 
they  are  widely  different  as  shown  in  Fig.  88,  the  fluidity  of 
the  cyclic  compounds  being  extraordinarily  low  even  at  their 
boiling  points,  marked  by  large  circles  in  the  figure.  The  higher 

engine  in  mid-air,  due  to  the  sudden  confining  of  the  gas  mixture  within 
the  cylinders  of  the  engine.  Flying  parts  of  the  engine  resulting  from  such 
an  explosion  may  also  injure  the  steering  mechanism,  the  supporting  planes, 
or  even  the  pilot. 


276 


FLUIDITY  AND  PLASTICITY 


fluidity  of  the  paraffin  is  strikingly  shown  by  introducing  a 
paraffin  residue  (CH3)  into  the  benzene  ring,  which  results  in 
toluene  (CeHg)  having  a  higher  fluidity  than  benzene  (C6H6). 
On  the  other  hand,  toluene  has  a  much  lower  fluidity  than  the 
purely  paraffin  compound  heptane  (C7Hi4)  which  contains  the 


800     900      1000 


300    400      500     600     700 
Vapor  Pressure  in  mm 

FIG.  89. — Fluidity-vapor-pressure  curves  of  hydrocarbons  of  different  homolo- 
gous series.     (Cf.  Fig.  58.) 

same  number  of  carbon  atoms.  It  may  be  urged  that  whereas 
these  compounds  contain  the  same  number  of  carbon  atoms  they 
do  not  contain  the  same  number  of  hydrogen  atoms.  But  one 
should  also  note  that  diallyl  contains  more  hydrogen  atoms  than 
benzene  and  less  than  hexamethylene  and  yet  has  a  fluidity 
which  is  far  higher  than  either.  The  cyclic  compounds  may  owe 
their  low  fluidity  to  association,  but  the  relation  of  association 
to  the  properties  desired  in  a  lubricant  is  not  well  understood. 
However,  the  relation  between  fluidity  and  vapor-pressure, 


LUBRICATION 


277 


already  discussed  (pages  155-160)  is  not  without  interest  in  this 
connection. 

Although  hexane,  diallyl,  benzene,  and  hexamethylene  differ 
in  fluidity  by  more  than  250  absolute  units  at  a  given  tempera- 
ture, they  all  boil  within  20  degrees  of  each  other,  hence  the 
fluidity-vapor  pressure  curves  for  these  hydrocarbons  are  very 
distinctive,  as  shown  in  Fig.  89.  If  a  low  vapor  pressure  for 
a  given  fluidity  is  an  advantage,  on  the  assumption  that  an  oil 
should  not  volatilize  off  from  the  walls  of  an  engine  cylinder  or 
away  from  an  overheated  bearing,  then  straight  chain  hydro- 
carbons have  the  apparent  advantage.  On  the  other  hand,  if 
low  vapor-pressure  and  high  molecular  weight  for  a  given  fluidity 
result  in  a  tendency  toward  carbonization,  then  cyclic  com- 
pounds will  be  preferred. 

TABLE  LXX. — AVERAGE   FLUIDITIES  AND  VAPOR  PRESSURES  FOR  CORRE- 
SPONDING TEMPERATURES 


Tem- 

Toluene1 

Benzene2 

Hexamethylene  3 

Hexane4 

pera- 

ture, 

Vapor 

Vapor 

Vapor 

Vapor 

degrees 

<f> 

pres- 

<f> 

pres- 

<p 

pres- 

•P 

pres- 

C 

sure 

sure 

sure 

sure 

0 

110.8 

25.3 

252.3 

45.4 

10 

131.5 

45.2 

84.7 

47.0 

281.8 

75.0 

20 

154.1 

75.6 

101.7 

76.9 

312.4 

120.0 

30 

192.3 

32 

178.0 

120.2 

121.3 

121.3 

344.8 

185.4 

40 

214.6 

60 

203.1 

183.6 

141.8 

181.6 

378.8 

276.7 

50 

238.6 

94 

228.8 

271.4 

164.8 

269.2 

414.9 

400.9 

60 

262.7 

140 

256.1 

390.1 

188.7 

385.0 

452.7 

566.2 

70 

287.8 

203 

284.9 

547.4 

214.5 

540.8 

494.6 

787.0 

80 

314.5 

291 

313.8 

751.9 

90 

342.8 

405 

100 

371.4 

560 

110 

400.0 

751 

1  Vapor  pressure  calculated  from  Kahlbaum,  Zeitschr.  f.  physik.  Chem. 
26,  603  (1898).     Fluidities  from  Bingham  and  Harrison,  Zeitschr.  f.  physik. 
Chem.  66,  1  (1909),  and  in  the  case  of  hexamethylene,  hitherto  unpublished 
data  of  Bingham  and  van  Klooster. 

2  From  Young,  J.  Chem.  Soc.  (London)  55,  486  (1889). 

3  From  Young  and  Fortney,  J.  Chem.  Soc.  (London)  75,  873  (1899). 

4  From  Thomas  and  Young,  J.  Chem.  Soc.  (London)  67,  1075  (1895). 


278  FLUIDITY  AND  PLASTICITY 

Anti-friction  Metals. — A  bearing  is  usually  made  of  a  different 
material  from  the  journal,  but  the  composition  of  the  so-called 
anti-friction  metals  varies  within  wide  limits.  It  must  be  soft 
enough  so  that  the  bearing  may  be  easily  scraped  and  quickly 
run  in  to  an  exact  fit.  During  the  process  of  "running  in"  the 
bearing,  the  particles  of  metal  doubtless  serve  to  wear  down  the 
high  spots  of  the  softer  metal,  leaving  the  harder  journal  in  a 
highly  polished  condition.  The  maximum  wear  is  naturally 
where  the  journal  and  bearing  are  in  closest  proximity,  hence  if  a 
bearing  has  been  run  in  with  motion  in  a  given  direction,  the 
coefficient  of  friction  will  be  altered  if  the  direction  of  motion  is 
reversed,  as  observed  by  Tower.  The  bearing  must  be  hard 
enough  to  carry  the  load  without  flow  of  the  metal.  It  seems 
probable  that  the  material  of  the  bearing  should  have  as  small 
an  adhesion  for  the  metal  of  the  journal  as  practicable  and  in  case 
of  necessity  the  material  of  the  bearing  should  be  capable  of 
acting  as  a  lubricant.  At  any  rate  it  should  not  tend  to  seize 
the  journal  even  when  molten.  Ice  may  be  regarded  as  the 
oldest  anti-friction  material,  and  from  certain  points  of  view 
it  is  ideal.  Since  it  melts  under  pressure,  it  furnishes  its  own 
lubricant  and  adhesion  does  not  occur  due  to  pressure.  A 
sleigh  standing  on  moist  ice  may  become  frozen  in  which  is 
evidence  that  adhesion  is  not  impossible  even  between  ice  and 
steel.  Adhesion  between  unlike  materials  is  less  serious  how- 
ever because  of  their  different  coefficients  of  expansion. 

Tin  is  a  common  constituent  of  anti-friction  metals  and 
there  was  a  serious  shortage  of  tin  during  the  late  war.  The 
lack  of  tin  ores  in  the  United  States  makes  very  desirable  the 
knowledge  of  alloys  which  do  not  contain  tin  and  at  the  same 
time  are  useful  for  bearings.  Experiments  indicate  that  lead 
containing  a  very  low  percentage  of  metallic  calcium  is  very 
satisfactory. 

Whether  the  alloy  should  have  a  certain  me-tajlographic 
structure,  as  for  example,  crystals  of  comparatively  hard  material 
imbedded  in  a  softer  amorphous  solid  is  a  moot  question. 


CHAPTER  XII 

FURTHER  APPLICATIONS  OF  THE  VISCOMETRIC 
METHOD 

There  are  many  further  applications  of  the  viscometric 
methods  which  are  destined  to  become  of  considerable  importance 
as  soon  as  the  theory  of  viscous  and  plastic  flow  is  thoroughly 
understood.  In  many  cases  however,  our  knowledge  at  present 
is  very  restricted,  or  it  is  the  closely  guarded  property  of  some 
industry.  Generally  speaking  however,  progress  has  been  held 
back  because  the  viscosity  data  at  hand  could  not  be  interpreted 
and  because  the  distinction  between  viscous  and  plastic  flow  was 
not  recognized.  An  illuminating  example  of  this  has  been 
described  by  Mr.  Gardner  and  Mr.  Ingalls.1  The  American 
Society  for  Testing  Materials  attempted  to  compare  with  all 
care  some  240  samples  of  paint,  applying  them  to  a  test  fence  at 
Arlington.  The  paints  were  all  made  up  to  have  the  same 
"viscosity"  as  measured  by  the  Stormer  viscometer.  Mr. 
Gardner  says  of  the  tests,  "The  determinations  were  fallacious. 
What  was  actually  done  was  to  make  some  paints  of  a  very  low 
and  some  of  a  very  high  yield  value,  although  they  all  measured 
up  to  the  same  viscosity.  The  result  was  that  when  some  of  the 
paints  were  applied  to  the  boards,  they  would  flow  and  carry  the 
pigment  particles  down,  leaving  bare  spots.  Some  paints 
failed  on  this  account." 

According  to  Batschinski's  Law  the  fluidity  varies  some 
2,000  times  as  rapidly  as  the  volume  which  is  now  used  success- 
fully in  the  dilatometric  method,  hence  the  viscometric  method 
should  be  most  useful  in  chemical  control  work.  Dunstan, 
Thole  and  their  coworkers  have  led  the  way  in  solving  chemical 
problems  by  means  of  viscosity  measurement.  They  have 
studied,  for  example,  the  keto-enol  tautomerism,  the  effect  of 
conjugate  bonds,  the  order  of  chemical  reactions  the  existence  of 

1  Proc.  A.  S.  T.  M.,  19,  Part  II,  (1919). 
279 


280  FLUIDITY  AND  PLASTICITY 

racemates  in  solution,  the  location  of  transition  points  such  as  the 
one  between  Na2SO4  and  Na2SO4.10H2O.1  Further  work  along 
this  line  is  needed  to  differentiate  the  effects  of  chemical  composi- 
tion, constitution,  and  association,  measuring  the  fluidities  over  a 
range  of  temperatures.  As  in  other  lines  of  physical  chemical 
investigation,  the  importance  of  making  determinations  at  more 
than  one  temperature  can  hardly  be  overestimated  because 
substances  must  be  compared  under  conditions  which  are  truly 
comparable. 

Various  colloidal  solutions  such  as  those  of  rubber,  glue,  vis- 
cose, nitrocellulose,  dextrine,  gluten,  et  cet.,  offer  problems  of 
importance  which  can  be  most  appropriately  solved  by  the  vis- 
cometer.  It  is  already  known  that  the  properties  of  a  solution 
of  caoutchouc,  for  example,  determine  the  character  of  the  rubber 
which  can  be  manufactured  from  it.  The  exact  relation  of  the 
viscosity  of  the  sol  to  the  plasticity  of  the  gel  is  practically  a 
closed  book.  To  indicate  how  complex  the  phenomena  may  be, 
we  may  add  that  Carl  Berquist  of  the  Corn  Products  Refining 
Company  has  found  in  an  investigation  of  corn  dextrines,  tapioca 
dextrine,  borax,  gums  and  starches  that  as  the  percentage  of 
dextrine  increases  during  the  process  of  conversion,  the  mobility 
steadily  rises  whereas  the  friction  first  falls,  then  rises,  and  again 
falls.2  The  quick  setting  of  a  gum  seems  to  be  associated  with  a 
high  friction.  Thus  the  addition  of  .25  per  cent  sodium  hydrox- 
ide to  a  8.33  per  cent  Pearl  starch  reduced  the  mobility  from 
0.7214  to  0.3018  but  increased  the  friction  from  108  to  156  g 
per  square  centimeter.  The  alkaline  starch  will  set  harder  and 
have  "better  body"  than  an  acid  starch. 

Nitrocellulose  Solutions. — The  fluidities  of  nitrocellulose  solu- 
tions as  calculated  from  the  determinations  of  Baker  (1913) 
would  indicate  that  nitrocellulose  solutions  never  become  true 
solids  as  the  percentage  of  nitrocellulose  is  increased,  for  the 
fluidities  approach  the  zero  value  asymptotically.  This  con- 
clusion is,  however,  so  inherently  improbable  that  it  should  be 
confirmed.  Since  it  was  necessary  to  use  a  series  of  Ostwald 
viscometers  in  order  to  get  the  necessary  range,  and  each  one  is 
calibrated  from  another,  the  possibility  of  error  is  considerable. 

lCf.  Dunstan  and  Langton   (1912). 

2  Privately  communicated.     Cf.  Herschel  and  Bergquist  (1921). 


APPLICATIONS  OF  THE  VISCOMETRIC  METHOD        281 

So  it  may  well  be  that  nitrocellulose  solutions  in  various  non- 
aqueous  solvents  may  be  brought  into  line  with  other  colloidal 
solutions,  some  of  which  have  already  been  considered,  page  198. 
If,  for  a  given  nitrocellulose,  there  is  a  zero  of  fluidity  which  is 
independent  of  the  particular  solvent,  an  empirical  formula  of 
the  general  type 


where  K  is  a  constant,  may  be  serviceable,  (cf.  Duclaux  and 
Wollman  (1920)). 

Colloidal  solutions  of  the  above  types  which  have  a  lattice- 
work or  sponge-like  structure  show  an  increase  in  the  fluidity 
when  subjected  to  treatment  which  breaks  up  this  structure. 
Astonishingly  small  quantities  of  the  disperse  phase  are  necessary 
to  give  zero  fluidity  or  at  any  rate  a  very  great  viscosity. 

Certain  non-polar  emulsion  colloids,  such  as  milk,  are  in  some- 
what sharp  contrast  with  the  above,  because  fairly  high  percent- 
ages of  the  disperse  phase  alter  comparatively  little  the  fluidity 
of  the  medium  and  the  reduction  of  the  size  of  the  fat  gluobles 
by  "homogenizing"  decreases  the  fluidity. 

Attempts  are  being  made  to  use  the  plasticity  method  in  the 
study  and  control  of  butter  and  other  fats  and  greases.  As  a 
means  for  distinguishing  between  different  fats  and  greases  and 
of  determining  the  amount  of  the  "hardening"  of  oils  in  the  proc- 
ess of  hydrogenation,  or  of  oxidation  in  the  blowing  of  oils,  the 
method  offers  opportunities  which  have  not  been  exploited  as 
yet.  Similarly  it  seems  practicable  to  estimate  the  amount  or 
quality  of  gluten  in  samples  of  flour  by  this  method. 

Clay  and  Lime.  —  Suspension  colloids  offer  a  simpler  set  of 
conditions  than  can  be  found  anywhere  else.  Clays,  plasters, 
mortars,  and  cements,  are  all  plastic  and  their  plasticity  is  a 
matter  of  prime  importance  in  their  respective  industries. 
Commenting  on  the  influence  the  plasticity  of  plaster  has  on  its 
economic  usefulness,  Emley  (1920)  states  that  about  70  per  cent 
of  the  total  cost  of  plastering  a  house  is  accounted  for  in  the 
labor  required  to  spread  the  plaster.  "If  one  plaster  is  more 
plastic  than  another,  it  means  that  the  plasterer  can  cover  more 
square  yards  in  a  given  time  with  the  former  than  with  the  latter, 
which,  of  course,  will  reduce  the  cost.  Furthermore  the  more 


282  FLUIDITY  AND  PLASTICITY 

plastic  material  entails  less  physical  and  mental  fatigue  on  the 
part  of  the  plasterer,  and  he  is  thereby  led  unwittingly  to  produce 
a  better  quality  of  work." 

Emley  points  out  that  the  method  of  slaking  the  lime  has  much 
to  do  with  the  development  of  plasticity,  but  that  quite  as  impor- 
tant is  the  source,  and  by  inference,  the  chemical  composition  of 
the  lime.  A  lime  high  in  magnesium  oxide  is  capable  of  develop- 
ing a  high  plasticity  more  readily  than  one  which  is  low  in  the 
dolomitic  oxide.  The  growing  practice  of  buying  Ohio  finishing 
lime,  already  hydrated,  even  when  local  lime  may  be  purchased 
for  about  one-half  the  price  is  a  reflection  of  the  above  facts  and 
is  a  demonstration  of  the  industrial  importance  of  plasticity. 

In  handling  road-building  and  roofing  materials,  a  knowledge 
of  the  principles  of  plastic  flow  might  enable  us  to  avoid  losses. 
The  first  principle  of  road  building  is  to  secure  proper  drainage, 
which  is  in  accord  with  the  theoretical  requirement  of  keeping 
the  yield  value  as  high  as  practicable.  The  "metal"  of  the  rail- 
road is  made  up  of  coarse  crushed  stone  of  uniform  size  which 
gives  excellent  drainage  and  a  very  high  yield  point.  Where 
liquid  hydrocarbons  are  used  as  binder,  a  considerable  amount  of 
fine  material  must  be  used  in  order  to  raise  the  yield  point  suffi- 
ciently to  sustain  the  contemplated  loads.  In  order  to  be  able 
to  apply  the  material  the  mobility  is  greatly  increased  by  raising 
the  temperature. 

Paints  and  Pigments. — Paint  must  have  a  yield  value  high 
enough  so  that  it  will  not  run  under  the  influence  of  gravity 
but  the  mobility  must  also  be  high  so  that  the  painter  may  spread 
it  without  undue  fatigue.  Other  things  being  equal,  these  ends 
are  both  achieved  by  the  use  of  finely-divided  materials,  and  at 
the  same  time  the  covering  power  is  augmented.  Perrott  (1919) 
has  made  a  study  of  the  plasticity  of  "long"  and  "short"  carbon 
blacks. 

Up  to  1914,  Austrian  ozokerite  was  thought  to  be  essential 
in  the  wax  used  in  making  electrotypes.  Research  has  shown 
that  a  good  impression  can  be  obtained  and  held  with  waxes 
which  do  not  contain  the  Austrian  material. 

Textiles  and  Belting. — If  a  cotton  window  cord  is  run  over  a 
free  pulley  a  certain  number  of  times  under  a  load  which  is  small 
in  comparison  with  the  tensile  strength  of  the  cord,  it  may  fail 


APPLICATIONS  OF  THE  VISCOMETRIC  METHOD       283 

while  another  cord,  apparently  no  better  as  judged  by  the  weight, 
tensile  strength,  method  of  fabrication  and  length  of  staple  will 
last  perhaps  one  hundred  times  as  long.  It  is  evident  that  oxi- 
dation or  decay  cannot  play  an  important  part  because  the  fail- 
ure may  be  brought  about  in  a  few  hours.  It  is  not  due  to 
friction  of  the  pulley  as  the  pulley  in  all  cases  is  running  free.  The 
surprising  thing  about  it  is  that  the  cord  often  wears  out  on  the 
side  which  is  away  from  the  pulley,  or  the  center  of  the  cord  may 
become  completely  pulverized  while  the  outside  is  apparently 
sound. 

An  analysis  of  what  happens  when  a  belt  moves  over  a  pulley 
shows  that  the  outside  of  the  belt  moves  along  a  longer  arc  and 
therefore  tends  to  get  ahead  of  the  inside  of  the  belt.  There  is 
consequently  a  shearing  stress  set  up  within  the  belt.  Since  the 
individual  fibers  are  comparatively  weak,  it  is  of  the  utmost 
importance  that  the  individual  fibers  be  protected  from  undue 
strains.  In  order  to  obtain  relief  where  the  strains  are  greatest, 
a  lubricant  between  the  fibers  and  plies  should  always  be  pro- 
vided. A  rosined  bow  adheres  to  a  violin  string  and  in  the  pro- 
duction of  sweet  sound  accumulates  stresses  advantageously, 
but  the  workman  who  gets  rosin  on  a  machine  belt  with  the  idea 
of  gaining  greater  traction,  may  quickly  bring  about  the  destruc- 
tion of  the  belt.  A  certain  amount  of  slipping  of  a  belt  and 
particularly  in  the  belt  is  necessary  and  desirable. 

Lard  and  certain  fixed  oils  are  used  to  "stuff"  leather,  and  a 
good  leather  belt  will  practically  never  wear  out  if  well-used  and 
dressed  with  lubricant  occasionally.  Window  cords  are  often 
lubricated  with  a  soft  paraffin.  The  paraffin  has  a  tendency  to 
work  out  in  use  and  since  it  becomes  hard  at  low  temperatures, 
it  then  tends  to  make  the  cord  stiff.  Pitch  and  its  congeners 
is  unsuitable  for  use  on  textile  belting  due  to  its  having  a  high 
temperature  coefficient  of  fluidity.  What  is  needed  as  a  lubricant 
is  a  substance  which  adheres  strongly  to  the  material,  lubricates 
the  fibers,  and  has  a  small  or  negligible  temperature  coefficient  of 
fluidity.  Oils  which  serve  well  with  leather  will  not  fill  the 
coarser  pores  of  textile  belting,  hence  rubber,  balata,  and  semi- 
drying  oils  are  often  used.  In  ordinary  fabrics  a  certain  amount 
of  oil  present  will  add  to  their  life.  Even  a  wire  rope  will  last 
longer  if  there  is  lubricant  between  the  strands. 


284  FLUIDITY  AND  PLASTICITY 

Metallurgy. — The  terms  hardness,  ductility,  pliability,  mallea- 
bility are  terms  which  are  probably,  like  the  term  plasticity, 
complex  in  character  and  may  in  time  come  to  be  more  precisely 
defined  in  terms  of  friction  and  mobility.  It  is  desirable  to 
know  the  friction  and  mobility  of  each  modification  of  each 
metal  and  their  several  alloys,  and  also  the  effect  upon  these 
properties  of  changes  in  crystal  size  or  shape  and  in  the  amount 
of  amorphous  solid  between  the  crystals.  This  subject  merits 
extended  treatment.  We  know  that  annealing  gives  the  crystals 
opportunity  to  develop  whereas  cold  working  tends  to  break  up 
the  crystal  structure  and  thereby  toughens  the  metal.  Quench- 
ing the  hot  metal  of  course  prevents  crystal  growth  and  should 
decrease  the  yield  point.  There  is  no  doubt  but  that  polishing 
and  similar  operations  result  in  a  plastic  flow  of  the  surface  layers 
of  a  metal. 

Biology,  Medicine  and  Pharmacy. — It  would  be  out  of  place 
here  to  treat  in  detail  of  the  very  numerous  papers  which  have 
been  devoted  to  biological  subjects.  Beginning  with  Poiseuille 
who  was  first  drawn  to  his  study  of  fluidity  through  his  interest 
in  the  circulation  of  the  blood  in  the  capillaries,  there  has  been  a 
continued  interest  in  the  viscosities  of  animal  liquids.  The 
viscosity  of  the  blood  in  various  individuals  and  species  of  animals, 
in  various  pathological  conditions  as  well  as  under  the  influence 
of  anaesthetics  and  drugs,  the  effect  on  viscosity  caused  by  differ- 
ences in  diet,  age,  sex,  or  temperature  outside  of  the  body,  the 
effect  upon  the  viscosity  of  the  blood  produced  by  the  removal  of 
certain  organs  of  the  body  have  all  been  subject  to  investigation. 
The  composite  character  of  the  blood  has  prompted  inquiries  in 
regard  to  the  viscosity  of  blood  serum  and  defibrinated  blood  as 
compared  either  with  blood  as  it  exists  within  the  animal  or  as  it 
is  freshly  drawn.  The  other  body  fluids,  milk,  lymph,  perspira- 
tion, the  vitreous  humor,  et  cet.,  have  all  been  studied  and  carefully 
reviewed  by  Rossi  (1906). 

Rossi  finds  that  preceding  the  coagulation  of  a  solution  there  is 
an  increase  in  viscosity  which  is  the  best  measure  of  the  progress 
toward  coagulation.  The  more  viscous  the  original  solution,  the 
more  rapidly  does  the  formation  of  the  gel  proceed.  Fano  and 
Rossi  (1904)  found  that  electrolytes  always  first  cause  a  drop  in 
the  viscosity  which  is  then  followed  by  a  rise  as  the  concentration 


APPLICATIONS  OF  THE  VISCOMETRIC  METHOD       285 

is  increased.  All  liquids  in  the  body,  whether  circulating  or  not 
have  the  minimum  viscosity  compatible  with  their  colloidal 
content. 

Oxygenated  blood  according  to  Haro  (1876)  is  much  more 
fluid  than  blood  through  which  carbon  dioxide  has  been  made  to 
bubble,  the  ratio  between  them  being  5.61  to  6.08.  Mere  phys- 
ical solutions  of  small  amounts  of  gases  in  liquids  usually  affect 
the  fluidity  but  imperceptibly,  but  the  data  on  this  subject  needs 
amplification.  Ether  and  ethyl  alcohol  added  to  the  blood 
increase  its  fluidity,  whereas  chloroform  has  the  opposite 
effect. 

Poiseuille  compared  the  rates  of  flow  of  blood  serum  through 
glass  tubes  and  through  a  given  vascular  territory  varying  the 
viscosity  of  the  serum  by  various  additions.  From  the  correla- 
tion it  has  been  assumed  that  the  laws  of  Poiseuille  apply  to  the 
flow  of  blood  through  the  capillaries  of  the  body.  Ewald  (1877) 
has  questioned  this  conclusion  and  Huebner  (1905)  has  noted  an 
incongruity  in  the  rate  of  flow  of  solutions  of  known  viscosity  in 
the  organs  of  a  frog.  When  blood  flows  through  the  capillaries 
the  corpuscles  are  deformed  and  the  capillaries  are  more  or  less 
elastic.  The  problems  connected  with  the  viscosity  of  the  blood 
are  complicated  by  the  fact  that  the  fibrinogen  of  the  blood  in 
contact  with  foreign  substances  produces  coagulation  which  may 
produce  a  coating  on  the  inside  of  the  tubes.  Lewy  (1897)  how- 
ever has  found  that  Poiseuille's  law  holds  good  so  long  as  no  sedi- 
mentation takes  place,  hence  the  more  viscous  the  blood  the 
longer  it  will  take  to  diffuse  through  a  given  vascular  territory. 

Burton-Opitz  (1914)  found  that  fasting  produced  a  pronounced 
increase  in  the  fluidity  of  the  blood  of  a  dog.  A  meat  diet  has 
the  greatest  effect  in  lowering  the  fluidity,  a  fat  diet  next,  and  a 
diet  of  carbohydrates  least  of  all.  The  fluidity  of  the  serum 
varies  in  a  manner  similar  to  that  of  the  blood  in  these  particular 
experiments. 

Bleeding  a  dog  causes  the  fluidity  of  the  blood  to  decrease. 
When  a  dog  was  kept  in  a  bath  at  43°C  the  fluidity  of  the  blood 
increased,  and  it  decreased  when  the  temperature  of  the  bath  was 
lowered  to  23°,  the  most  rapid  change  taking  place  in  from  5  to 
15  minutes  according  to  Huerthle  (1900). 

According  to  Huebner  the  red  blood  corpuscles  account  for 


286  FLUIDITY  AND  PLASTICITY 

from  two-thirds  to  three-quarters  of  the  viscosity  of  the  blood. 
The  fluidity  of  the  blood  of  cold-blooded  animals  is  higher  than 
that  of  warm-blooded  animals,  but  the  rabbit  is  peculiar  among 
warm-blooded  animals  in  having  blood  of  exceptionally  high 
fluidity. 

Milk. — The  fluidity  of  the  milk  of  a  cow  differs  from  day  to 
day  as  well  as  with  different  individuals  and  at  different  periods 
of  life.  The  fluidity  of  woman's  milk  is  highest  directly  after 
childbirth  and  falls  off  nearly  50  per  cent  during  the  period  of 
nursing.  The  milk  of  goats  is  less  fluid  than  that  of  cows. 
According  to  Cavazzani  (1905)  the  addition  to  milk  of  small 
amounts  of  NaOH  or  KOH  produces  a  change  in  the  fluidity  of 
the  milk  of  a  cow,  goat,  or  horse  but  does  not  affect  the  fluidity  of 
woman's  milk.  According  to  Alexander1  human  milk  contains 
a  protective  colloid  not  present  in  cow's  milk,  hence  coarse  curds 
are  not  formed  on  adding  acids. 

The  action  of  ferments  upon  milk  has  been  studied  by  Gutzeit 
(1895)  and  Fuld  (1902).  The  decline  in  the  viscosity  of  a  solu- 
tion of  proteins  during  digestion  by  means  of  trypsin  has  been 
the  subject  of  study  by  Spriggs  (1902).  The  greater  part  of  the 
loss  in  viscosity  occurs  considerably  before  the  completion  of 
the  digestion,  according  to  Bayliss  (1904).  This  is  in  accordance 
with  the  idea  that  the  destruction  of  the  structure  must  lower  the 
viscosity  tremendously,  whereas  the  splitting  of  microscopic 
particles  may  increase  the  viscosity  and  the  splitting  of  amicro- 
scopic  particles  decreases  the  viscosity.  Spriggs  (1902)  and 
Zanda  (1911)  investigated  the  changes  in  viscosity  during  diges- 
tion by  pepsin. 

Ceramics  and  Glass  Making. — The  thorough  mixing  of  glass 
melts,  the  removal  of  bubbles  of  gases,  and  the  pressure  necessary 
to  blow  the  glass  at  a  given  temperature  all  depend  upon  the 
fluidity  of  the  melt,  hence  the  control  of  the  fluidity  of  glass 
melts  is  of  importance. 

The  Seger  cone  method  of  determining  temperatures  suggests 
the  possibility  of  measuring  high  temperatures  by  the  viscometric 
method.  Barus  proposed  to  use  the  viscosity  of  a  gas  for  this 
purpose. 

The  manufacture  of  porcelain  is  concerned  with  the  principles 

1 J.  Soc.  Chem.  Ind.  28,  280  (1909). 


APPLICATIONS  OF  THE  VISCOMETRIC  METHOD       287 

of  plastic  flow  at  every  stage.  Clays  must  have  a  friction  high 
enough  so  that  the  ware  will  not  lose  its  shape  while  in  the  moist 
condition  and  at  the  same  time  it  must  have  a  mobility  which  is 
high  enough  so  that  the  clay  may  be  readily  worked  and  it  must 
not  shrink  badly  on  drying.  On  heating,  the  more  fusible  parti- 
cles must  soften  sufficiently  to  weld  the  particles  together,  but 
again  the  friction  must  be  sufficient  so  that  there  will  be  no  serious 
loss  of  shape.  When  the  glaze  is  added,  it  must  fill  the  pores 
quickly  and  yet  not  "run."  So  many  problems  in  plastic  flow 
seem  to  call  for  precise  control  of  conditions  in  order  to  avoid 
large  losses. 

It  is  found  that  considerable  amounts  of  non-plastic  clay,  fine 
sand,  or  ground  porcelain  (grog)  may  be  added  to  a  very  plastic 
clay  without  greatly  lowering  its  plasticity.  Until  more  data 
is  accumulated,  this  may  remain  something  of  a  mystery,  but 
these  additions  are  valuable  and  probably  serve  somewhat  the 
function  of  the  "reinforcing"  in  concrete  or  of  the  colloid  in 
"solidified  alcohol." 

Geo -physics. — Basic  lavas  are  notably  fluid  as  compared  with 
acidic  lavas  which  are  more  viscous.  This  has  important  bear- 
ings upon  the  character  of  volcanic  eruptions  in  different  parts 
of  the  world  and  presumably  therefore  upon  the  past  history  of 
the  earth.  For  example,  the  Hawaiian  volcanoes  with  a  highly 
basic  lava  tend  to  remain  open,  flow  quietly,  build  a  low-angle 
cone,  the  lava  spreading  out  over  a  large  amount  of  territory.  On 
the  other  hand,  the  Mexican  volcanoes  with  acidic  lava  are  apt 
to  harden  over  during  quiescence  and  then  erupt  violently.  A 
low-angle  cone  is  impossible.  In  accordance  with  the  relationship 
between  the  fluidity  of  the  melt  and  the  rate  of  crystallization, 
we  should  expect  to  find  the  basic  lavas  more  coarsely  crystalline 
than  those  of  a  more  acidic  nature.  The  length  of  time  required 
for  an  obsidian  to  take  on  a  cryptocrystalline,  microcrystal- 
line  or  even  macrocrystalline  character  will  of  course  also  de- 
pend upon  the  temperature  and  to  a  lesser  extent  upon  the 
pressure  as  well  as  the  chemical  composition,  for  all  of  these 
factors  influence  the  fluidity.  Silicate  melts  have  been  studied  by 
Doelter  (1906). 

Segregations,  as  in  the  separation  of  iron  from  slag,  is  depend- 
ent to  a  certain  extent  upon  the  fluidity  of  the  slag  and  of  the 


288  FLUIDITY  AND  PLASTICITY 

molten  metal.     Feild   (1918)   has  investigated  the  viscosity  of 


The  sodium  silicate  used  in  industry  contains  varying  hydroxyl 
ion  concentration.  An  excess  of  silicic  acid  increases  the  adhes- 
iveness but  lowers  the  mobility.  Excess  of  alkali  has  the  opposite 
effect.  The  alkalinity  of  sodium  silicate  is  therefore  obviously 
an  important  control  factor. 

Conclusion. — If  one  plots  the  viscosity-concentration  curves 
of  a  colloid  sol  of  the  type  of  gelatine  in  water  or  of  nitrocellulose 
in  acetone,  one  finds  that  the  viscosity  rapidly  goes  from  the 
very  small  viscosity  of  the  pure  solvent  (O.Olp  for  water  at  20° 
and  0.003  for  acetone  at  25°)  to  an  extremely  high  value  which 
may  be  regarded  as  infinite,  in  a  concentration  of  only  a  few  per 
cent.  Plotting  these  curves  leads  to  unsatisfactory  results, 
which  need  not  be  exhibited  here  as  they  are  very  common  in  the 
literature;  the  curves  fall  together  at  one  extreme  as  soon  as  one 
tries  to  represent  more  than  the  most  dilute  solutions,  and  where- 
as they  may  or  may  not  coincide  at  the  other  extreme,  we  can 
form  no  idea  of  what  happens  since  that  extreme  is  infinitely 
removed  from  us. 

If  however  we  plot  fluidities  instead  of  viscosities  the  whole 
problem  becomes  immediately  simplified,  for  the  fluidities  of  the 
pure  solvents  assume  their  proper  importance  and  the  fluidity 
goes  to  or,  at  any  rate,  approaches  zero,  which  is  accessible. 
Moreover  the  concentration  of  zero  fluidity  has  a  definite  and 
important  significance.  If  the  relation  turns  out  to  be  also 
linear,  then  the  problem  is  one  of  ideal  simplicity. 

To  go  over  all  of  the  data  in  the  literature,  critically  examining 
the  data  to  see  how  far  it  could  be  used  to  support  and  further 
amplify  the  theories  set  forth  in  this  work  has  been  a  pleasant 
task  but  far  too  great  for  a  single  worker.  Already  several 
workers  are  in  the  field  and  in  the  Index  and  Appendix  we  are 
bringing  together  a  considerable  number  of  references  and  tables 
in  order  to  facilitate  the  work. 

A  consideration  of  the  following  data  may  aid  any  who  are 
interested  in  the  theoretical  study  of  colloids  or  in  their  industrial 
applications,  since  they  help  us  to  answer  the  very  important  and 
novel  questions:  "Are  fluidity-temperature  curves  linear  in  the 
case  of  emulsoid  colloids  of  the  type  of  gelatine?"  "Are  their 


APPLICATIONS  OF  THE  VISCOMETRIC  METHOD        289 


fluidity-concentration  curves  linear?"  and  "Does  the  fluid  'sol' 
pass  into  the  plastic  'gel'  at  a  perfectly  definite  concentration  and 
temperature?" 

Arisz  (1915)  has  made  a  valuable  study  of  the  viscosity  of  a 
10  per  cent  gelatine  sol  in  a  glycerol-water  mixture  of  1.175 
specific  gravity,  with  changing  temperatures.  Calculating  the 
fluidities,  we  obtain  the  linear  curve 


0.000227  (t  -  45.2) 


(112) 


which  represents  faithfully  the  observed  values  of  the  fluidities 
given  in  Table  LXXI.  The  temperature  of  zero  fluidity  is  a 
little  over  45°,  which  must  therefore  be  regarded  as  the  transition 
point  between  the  fluid  and  solid  phases,  i.e.,  the  melting-point. 

TABLE  LXXI. — THE  FLUIDITIES  OF  10  PER  CENT  GELATINE  SOLUTION  IN 
GLYCEROL-WATER  MIXTURE  OF  DIFFERENT  TEMPERATURES  (AFTER  ARISZ) 


Temperature, 
degrees 

Viscosity, 
observed 

Fluidity, 
observed 

Fluidity, 
calculated 

44 

30,000 

0.000003 

0.0000 

46 

5,000  ± 

0.00020+ 

0.0002 

47 

4,200- 

0.00024  + 

0.0004 

50 

950  ± 

0.00105  + 

0.0011 

55 

415 

0.0024 

0.0022 

65 

222 

0.0045 

0.0045 

In  considering  the  effect  of  concentration  on  fluidity,  we  cite 
first  the  data  of  Lliers  and  Schneider  (1920)  on  flour-water  mix- 
tures for  20°,  giving  the  concentrations  in  volume  percentages 
(1006)  and  changing  the  viscosities  to  fluidities,  Table  LXXII. 
The  fluidities  are  again  faithfully  reproduced  by  means  of  a 
linear  formula,  viz., 

<p  =  100.5  -  569.66  (113) 

except  the  last  two  concentrations,  where  the  observed  fluidity 
is  too  high.  This  gives  a  concentration  of  zero  fluidity  as  17.6 
per  cent,  which  corresponds  to  the  transition  from  viscous  to 
plastic  flow.  This  is  very  close  to  the  concentrations  of  zero 
fluidity  found  by  Bingham  (1916)  for  clay  suspensions,  page  229. 


290 


FLUIDITY  AND  PLASTICITY 


TABLE  LXXII. — THE  FLUIDITIES  OF  VARIOUS  VOLUME  CONCENTRATIONS 
OF    MALT   FLOUR-WATER    MIXTURES   AT   20°C    (AFTER   LUERS   AND 

SCHNEIDER) 
1  g  flour  displaces  0.6766  ml  of  toluene 


Volume,  per  cent 

Viscosity  relative, 
observed 

Fluidity  absolute, 
observed 

Fluidity  absolute, 
calculated 

0.000 

1.000 

100.5 

100.5 

0.338 

1.010 

99.5 

98.6 

0.676 

1.042 

96.5 

96.6 

1.352 

1.090 

92.2 

92.8 

2.704 

1.192 

84.3 

85.1 

5.408 

1.433 

70.1 

69.7 

8.120 

1.844 

54.5 

54.2 

10.816 

2.566 

39.2 

38.9 

13  .  520 

3.459 

29.0 

23.5 

16.240 

4.839 

20.8 

8.0 

There  are  certain  cases  where  the  fluidity-concentration 
relation  is  not  linear  as  in  case  of  Baker's  data  for  nitrocellulose 
solutions.  It  seems  unwise  to  make  any  sweeping  deductions 
in  regard  to  the  meaning  of  these  curves  until  they  are  confirmed 
by  further  observations,  for  it  is  possible  that  correction  terms 
applied  to  the  measurements  might  serve  to  rectify  the  curves. 
It  is  perhaps  needless  to  add  that  if  the  curvature  is  real,  it  will 
have  an  important  bearing  upon  the  nature  of  colloids.  I  am 
engaged  upon  a  study  of  this  whole  matter  at  the  present  time. 

In  a  few  cases,  of  which  one  has  been  already  cited,  page  207, 
the  fluidity  curve  consists  of  two  linear  branches,  meeting  at 
a  sharp  angle.  This  would  seem  to  indicate  a  transition  point, 
but  in  view  of  the  uncertainties  connected  with  the  measurements 
of  the  viscosities  of  colloidal  solutions,  we  may  be  pardoned  for 
extreme  caution  in  making  such  an  assumption  until  the  real 
existence  of  such  a  singular  point  has  been  thoroughly  verified. 
Such  a  singular  point  is  found  for  example  in  the  data  for  aqueous 
solutions  of  sodium  palmitate  at  70°C  as  determined  by  Farrow 
(1912).  The  formula 

v  =  240 .4  -  5Q3.9bw  (114) 

represents   the  relation  between  the    fluidity  and  the  weight 


APPLICATIONS  OF  THE  VISCOMETRIC  METHOD       291 


concentration  up  to  a  concentration  of  0.175  but  this  and  the 
remaining  observations  are  reproduced  by  the  formula 

<f  =  207.7  -  306.76*.  (115) 

Neither  formula  gives  the  true  fluidity  of  water  at  70°  (245.7) 
when  the  weight  concentration  bw  of  palmitate  is  zero,  and  we 
obtain  two  different  concentrations  corresponding  to  zero  fluidity, 
using  the  two  formulas,  of  0.477  and  0.677  respectively,  which  are 
difficult  to  interpret. 

TABLE  LXXIII. — FLUIDITIES  OF  AQUEOUS  SOLUTIONS  OF  SODIUM  PALMITATE 
AT  70°C  (AFTER  FARROW) 


Concentration  by 
weight 

Viscosity,    ob- 
served 

Fluidity,  observed 

Fluidity,  calcu- 
lated 

0.026 

0.00438 

228 

227 

0.064 

0.00485 

206 

208 

0.105 
0.131 

0.00530 
0.00579 

189 
173 

188 
174 

Formula  I 

0.141 

0.00592 

169 

169 

0.175 

0  .  00647 

155 

152 

0.175 

0.00647 

155 

154 

0.187 

0.00665 

150 

150 

0.232 

0.00743 

135 

136 

0.283 

0.00793 

126 

121 

0.287 

0.00866 

115 

120 

0.293 

0.00847 

118 

118 

Formula  II 

0.302 

0.00886 

113 

115 

0.329 

0.00949 

105 

107 

0.380 

0.01113 

90 

91 

0.451 

0.01363 

73 

69 

0.499 

0.01800 

56 

55 

W.  L.  Hyden  has  made  a  study  of  the  plasticity  of  solutions 
of  nitrocellulose  in  acetone,  but  the  results  have  not  yet  been 
published.  He  finds  that  these  colloidal  solutions  differ  from 
the  suspensions  of  clay  et  cet.  studied  by  Durham  in  that  the 
concentration  of  zero  fluidity  at  a  very  low  shear,  i.e.,  the  transi- 
tion from  viscous  to  plastic  flow,  occurs  in  an  extraordinarily  low 
concentration  of  colloid,  certainly  less  than  one  per  cent. 

Since  the  apparent  viscosity  of  such  colloid  solutions  is  depen- 


292 


FLUIDITY  AND  PLASTICITY 


dent  upon  the  amount  of  the  shearing  force,  the  values  of  the 
viscosity  as  such  are  quite  illusory.  For  example,  a  1.39  per 
cent  solution  of  nitrocellulose  in  acetone  gave  a  fluidity  of 
52.49  using  a  pressure  of  403.6  g  per  cm2  whereas  the  fluidity 
apparently  fell  to  51.31  at  214.5  g  per  cm2  and  to  50.76  at  62.96 
g  per  cm2.  These  results  are  similar  to  those  of  Glaser  (cf. 
Table  XVIII). 

It  is  quite  possible  to  measure  the  plasticity  of  materials  of 
this  kind  in  the  viscometer  shown  in  Fig.  29.  It  is  merely 
necessary  to  measure  the  flow  at  two  or  more  pressures  and  then 


50 


40 


c  30 
o 


n-  20 


3  4  5 

Concentration 

FIG.  90. — Friction-weight^concentration     curve     for    colloidal    dispersions    of 
nitrocellulose  in  acetone  measured  in  dynes  per  square  centimeter. 

compare  the  volume  of  flow  with  the  shearing  stress.  The 
volume  of  flow-shearing  stress  curves  obtained  in  this  way  by 
Hyden  are  linear  in  every  case.  The  mobility  is  found  to  in- 
crease with  the  temperature  in  a  nearly  linear  manner  and  the 
mobility  falls  off  very  rapidly  with  increasing  concentration  of 
colloid,  approaching  the  zero  value  asymptotically.  Both  of  these 
results  are  similar  to  those  for  clay  suspensions  (cf.  pages  220 
and  221).  The  equations  of  these  curves  have  not  yet  been 
obtained. 

The  friction  in  nitrocellulose  solutions  increases  rapidly  with 
increasing  concentration  of  colloid,  as  shown  in  Fig.  90.  As  the 
temperature  is  raised  the  friction  is  decreased  in  a  linear  manner, 


APPLICATIONS  OF  THE  VISCOMETRIC  METHOD        293 

Fig.  91,  so  that  at  about  43  °C  nitrocellulose  in  acetone  would 
appear  to  have  the  properties  of  a  true  fluid.  If  above  43°  we 
have  a  true  solution,  this-  temperature  is  a  transmission  point 
which  is  analogous  to  the  melting  point  of  a  solid.  This  does 
not  mean  however  that  there  would  be  any  marked  change  in 
the  working  properties  as  nitrocellulose  solutions  below  43°  are 
extremely  soft  solids. 


0 

50 
40 

30C 

20 

o 
10 

n° 

•s. 

\ 

^ 

\ 

^\^ 

~^*v 

\ 

\. 

7*N>< 

^ 

'0  10  20  30  40  50  60  70 

Friction  in  dynes  per  crn2 

FIG.  91. — Temperature-friction  curve  for  a  colloidal  dispersion  of  7. 70S  weight 
percentage  of  nitrocellulose  in  acetone. 

Solubility  and  Plasticity. — It  is  of  course  well-known  that  the 
so-called  "solutions"  of  nitrocellulose,  gelatine  and  other  colloids 
are  not  true  solutions,  nevertheless  the  term  solution  as  applied 
to  colloidal  dispersions  often  leads  to  confusion.  Thus  acetone 
is  one  of  the  best  "solvents"  for  nitrocellulose,  being  superior 
to  let  us  say,  amyl  acetate.  But  what  does  this  statement 
mean?  It  cannot  possibly  mean  that  acetone  will  actually 
dissolve  more  nitrocellulose  than  will  a  similar  amount  of  amyl 
acetate,  for  there  is  no  point  of  saturation  for  either,  i.e.,  both 
liquids  will  " dissolve"  or  better  disperse  an  indefinite  amount  of 
colloid;  hence  the  term  solubility  has  here  a  very  special,  albeit 
a  very  definite,  meaning,  viz.,  that  dispersive  medium  is  the  best 
solvent  which  with  a  given  amount  of  colloid  gives  an  emulsion 
having  the  maximum  mobility.  Here  however  there  enters  the 


294  FLUIDITY  AND  PLASTICITY 

fact,  which  seems  from  the  literature  not  to  have  been  sufficiently 
considered,  that  acetone  has  a  far  greater  fluidity  than  amyl 
acetate  to  start  with,  and  this  must  of  necessity  affect  the 
mobility  of  dispersions  in  these  media.  It  is  evident  that  this 
must  be  taken  into  account  if  we  are  to  get  a  true  measure  of  the 
dispersive  power  of  different  media.  This  work  is  being 
continued. 


APPENDIX  A 
PRACTICAL  VISCOMETRY 

The  most  essential  part  of  the  viscometer  is  shown  in  Fig.  29 ,  p .  76 . 
To  use  the  apparatus  an  appropriate  amount  of  the  liquid  whose 
viscosity  is  to  be  measured  is  pipetted  into  the  right  limb.  The 
liquid  at  the  desired  temperature  is  forced  over  into  the  left 
limb  until  the  right  meniscus  reaches  the  point  N,  it  being  noted 
that  there  is  sufficient  liquid  so  that  the  surplus  runs  over  into 
the  trap.  The  right  limb  is  turned  to  air  so  as  to  prevent  more 
liquid  from  flowing  into  the  trap.  Having  adjusted  the  working 
volume,  the  left  limb  is  connected  with  the  pressure,  and  the  time 
required  for  the  left  meniscus  to  fall  from  B  to  D  is  noted.  The 
left  limb  is  now  turned  to  atmospheric  pressure  and  the  instru- 
ment is  ready  for  an  immediate  duplicate  determination  in  the 
opposite  direction.  In  this  second  measurement  the  time  is 
noted  which  is  required  for  the  left  meniscus  to  rise  from  D  to  B. 

Knowing  the  pressure,  p  in  grams  per  square  centimeter,  the 
time,  t,  in  seconds,  the  two  constants  of  the  instrument,  C  and  C", 
and  the  density  of  the  liquid,  p,  the  viscosity  77  at  the  given  tem- 
perature is  given  by  the  formula,  (cf.  p.  74). 

17  =  Cpt  -  C'p/t  (1) 

DETERMINATION  OF  THE  CONSTANTS  OF  THE  INSTRUMENT 

The  second  term  of  the  right  hand  member  of  the  above  equa- 
tion is  the  kinetic  energy  correction  which  should  never  exceed 
5  per  cent  of  the  value  of  the  first  term.  For  this  reason  the 
value  of  the  constant  C'  and  of  the  density  p  need  be  known  with 
an  accuracy  of  2  per  cent  only  in  order  to  allow  viscosity  deter- 
minations to  be  made  with  an  error  of  only  one-tenth  of  1  per 
cent. 

C'  =  0.0446  V/l  (2) 

where  V  is  the  volume  in  milliliters  of  the  bulb  C  between 
the  marks  B  and  D,  and  Z  is  the  length  of  the  capillary  EF, 
log.  0.0446  =  8.64895  -  10. 

295 


296  FLUIDITY  AND  PLASTICITY 

The  value  of  the  constant  C  is  most  conveniently  obtained  by 
filling  the  instrument  with  freshly  distilled,  dust-free  water  and 
determining  the  time  of  flow  for  each  limb,  at  20°C. 
0.01005*  +  C'p 

tt^2 W 

for  water  at  20°.  This  constant  may  also  be  obtained  by  direct 
measurement 

C  =  384.8r4/FZ  (4) 

where  r  is  the  radius  of  the  capillary  in  centimeters.  If  the 
acceleration  of  gravitation  of  the  locality  is  not  980,  the  value 
of  C  must  be  increased  0.1  per  cent  for  each  unit  in  excess. 

Since  the  bulbs  C  and  K  may  differ  in  level,  it  is  evident  that 
the  pressure,  p,  used  in  calculating  the  viscosity  is  not  necessarily 
equal  to  the  pressure,  pi}  delivered  by  the  compressed  air  at  the 
top  of  the  viscometer.  If  the  bulb  K  is  higher  than  the  bulb  C 
by  a  distance  h,  then  it  is  evident  that  the  pressure  during  the  left 
limb  determination  is  decreased  by  an  amount  htp  and  the  pres- 
sure during  the  right  limb  determination  is  increased  by  the  same 
amount.  Hence, 

Pi  -  /ZIP  =  ^  +c'tP^~  ^left  limb) 
?>2  +  hlP  =  *  +%'i'/t*  (right  limb) 
and  therefore 

hl  =  2Cp  \ti      tj  +  2C  Ua  ~~  iff  "    ~2~ 

or  if  the  two  determinations  are  carried  out  with  water  at  20° 
using  the  same  pressure 
0.005034 


where  t,  is  the  time  of  flow  from  right  to  left  and  fa  is  the  time  of 
flow  from  left  to  right.  Log.  0.005034  =  7.70191  -  10. 

The  value  of  C  used  in  calculating  the  hydrostatic  head  is  an 
approximate  value  obtained  from  Eq.  (3)  by  employing  the  pres- 
sure, pi,  uncorrected  for  hydrostatic  head,  which  is  legitimate 
since  the  hydrostatic  head  is  at  the  most  only  a  small  correction 
term. 

J.  W.  Temple  has  worked  out  a  simpler  method  for  calculating 


APPENDIX  A  297 

the  hydrostatic  head  when  the  flow  in  opposite  directions  is 
carried  out  at  the  same  manometer  pressure  p.  Let  the  time  of 
flow  in  the  one  direction  tL,  under  the  true  pressure  corrected  for 
hydrostatic  head  pL  =  p  +  hip,  be  supposed  to  be  less  than  the 
time  tR  in  the  opposite  direction  under  the  pressure  pR  =  p 
-  h1P.  Then 

,  PL  ~  PR 

hi  =  —  s  — 
2p 

and  substituting  into  this  equation  the  values  of  pL  and  pR  given 
by  Eq.  (1),  we  have 

1  h  +  C'p/tL       r,  +  C'p/tR\ 
2p\       CtL  CtB       I 

but  in  the  kinetic  energy  correction,  which  is  itself  always  small, 
the  small  hydrostatic  head  correction  is  of  negligible  influence, 
hence  for  our  purpose  we  may  write  rj  -f-  C'p/tL  =  rj  +  C'p/tR  so 


but  from  Eq.  (1)  we  have  that  T?  +  C'p/tL  =  CpLtL 
hence 


«> 


THE  TRUE  AVERAGE  PRESSURE 

It  might  inadvertently  be  assumed  that  if  the  two  bulbs  C  and 
K  are  the  same  in  shape  and  volume  and  also  at  the  same  level, 
the  true  pressure  to  be  used  in  calculating  the  viscosity  would 
necessarily  be  the  pressure  p\  delivered  by  the  compressed  air  in 
the  viscometer  because  the  hydrostatic  head  as  obtained  above 
would  then  be  zero.  But  since  the  hydrostatic  head  in  the  vis- 
cometer is  really  continually  changing,  the  true  average  pressure 
may  not  be  zero  under  the  above  conditions,  and  it  must  be 
obtained  by  integration.  Bingham,  Schlesinger,  and  Coleman 
(1916)  1  have  shown  that  for  cylindrical  bulbs  the  true  average 
pressure  p  would  be 


1  For  other  shapes  of  bulbs  see  original  paper  of  Bingham,  Schlesinger, 
and  Coleman.  For  the  possible  importance  of  such  corrections  see  Kendall 
and  Munroe  (1917). 


298 


FLUIDITY  AND  PLASTICITY 


where  h  is  the  height  of  the  bulb  and  po  is  the  pressure  with  all 
other  corrections  made.  Fortunately  if  the  height  of  the  bulb 
BD,  in  Fig  29,  is  not  more  than  one- thirtieth  of  the  whole  pressure, 
this  correction  is  unnecessary  to  attain  the  desired  accuracy  of 
0.1  per  cent. 

In  any  case,  however,  the  student  should  determine  by  experi- 
ment whether  a  change  in  manometer  pressure  is  without  effect 
upon  the  valve  of  C. 

THE  PRESSURE  CORRECTIONS  OUTSIDE  THE  VISCOMETER 

Let  the  density  of  the  liquid  within  the  manometer  be  p0  at  a 
temperature  T  in  degrees  Centigrade  and  the  height  read  on  the 
manometer  scale — corrected  for  scale  error  if  necessary — be  h0; 
also  let  the  viscometer  bulbs  be  at  a  height  h'  above  the  middle 
point  of  the  manometer.  The  pressure  delivered  to  the  air  in 
the  viscometer  becomes 

Pi  =  h0  —  K  ±  L  for  a  water  manometer  (8) 

Pi  =  M  ±  N  for  a  mercury  manometer  (9) 

where  the  values  of  L  are  given  in  Table  I  and  may  be  made 
entirely  negligible  in  the  setting  up  of  the  apparatus. 

TABLE  I.— VALUES  OF  L 


ho  in  centimeters 

100 

200 

300 

50 

0.01 

0.01 

Q.02 

100 

0.01 

0.03 

0.04 

•200 

0.03 

0.05 

0.08 

300 

0.04 

0.08 

0.11 

APPENDIX  A 

The  values  of  K  are  given  in  Table  II. 
TABLE  II. — VALUES  OF  K 


299 


Temperature, 
degrees  centi- 
grade 

Manometer    reading,    ho 

10 

20 

30 

40 

50 

60 

70 

80 

90 

100 

200 

300 

5 

0.0130.025 

0.039 

0.053 

0.066 

0.079 

0.094 

0.108 

0.122 

0.136 

0.285 

0.482 

10 

0.016 

0.030 

0.046 

0.064 

0.078 

0.095 

0.112 

0.129 

0.145 

0.162 

0.337 

0.533 

11 

0.017 

0.032 

0.050 

0.068 

0.083 

0.101 

0.119 

0.137 

0.154 

0.172 

0.357 

0.563 

12 

0.018 

0.035 

0.053 

0.072 

0.089 

0.108 

0.126 

0.145 

0.163 

0.183 

0.379 

0.596 

13 

0.019 

0.037 

0.057 

0.077 

0.095 

0.115 

0.135 

0.155 

0.175 

0.195 

0.403 

0.632 

14 

0.020 

0.040 

0.061 

0.082 

0.102 

0.123 

0.144 

0.165 

0.187 

0.208 

0.429 

0.671 

15 

0.022 

0.043 

0.065 

0.088 

0.110 

0.131 

0.154 

0.177 

0.199 

0.222 

0.457 

0.713 

16 

0.023 

0.046 

0.070 

0.094 

0.118 

0.140 

0.165J0.189 

0.212 

0.238 

0.489 

0.761 

17 

0.025 

0.049 

0.075 

0.101 

0.126 

0.150 

0.17610.203 

0.228 

0.255 

0.523 

0.812 

18 

0.027 

0.053 

0.080 

0.108 

0.135 

0.161 

0.189J0.217 

0.245 

0.273 

0.559 

0.866 

19 

0.029 

0.057 

0.086 

0.116 

0.144 

0.173 

0.2030.233 

0.262 

0.292 

0.597 

0.923 

20 

0.031 

0.060 

0.092 

0.124 

0.154 

0.185 

0.2170.249 

0.280 

0.312 

0.637 

0.983 

21 

0.033 

0.065 

0.098 

0.132 

0.165 

0.198 

0.2320.265 

0.299 

0.333 

0.679 

1.046 

22 

0.035 

0.069 

0.105 

0.141 

0.176 

0.211 

0.2470.282 

0.319 

0.355 

0.723 

1.113 

23 

0.037 

0.074 

0.112 

0.151 

0.188 

0.225 

0.264 

0.301 

0.341 

0.379 

0.770 

1.184 

24 

0.040 

0.079 

0.119 

0.160 

0.200 

0.240 

0.281 

0.321 

0.363 

0.403 

0.819 

1.256 

25 

0.042 

0.084 

0.127 

0.170 

0.212 

0.255 

0.298 

0.341 

0.385 

0.428 

0.869 

1.331 

26 

0.045 

0.089 

0.135 

0.181 

0.225 

0.270 

0.316 

0.362 

0.408 

0.454 

0.921 

1.409 

27 

0.048 

0.094 

0.143 

0.191 

0.239 

0.286 

0.335 

0.383 

0.432 

0.481 

0.975 

1.490 

28 

0.050 

0.100 

0.151 

0.202 

0.253 

0.303 

0.355 

0.405 

0.458 

0.509 

1.031 

1.574 

29 

0.053 

0.105 

0.160 

0.214 

0.268 

0.321 

0.375 

0.429 

0.484 

0.538 

1.089 

1.661 

30 

0.056 

0.111 

0.169 

0.226 

0.283 

0.339 

0.396 

0.453 

0.511 

0.568 

1.149 

1.751 

31 

0.059 

0.117 

0.178 

0.239 

0.298 

0.357 

0.417 

0.478 

0.538 

0.599 

1.210 

1.842 

32 

0.062 

0.124 

0.188 

0.251 

0.314 

0.376 

0.439 

0.503 

0.567 

0.630 

1.273 

1.937 

33 

0.066 

0.130 

0.197 

0.264 

0.330 

0.395 

0.462 

0.529 

0.595 

0.662 

1.337 

2.033 

34 

0.069 

0.137 

0.207 

0.277 

0.346 

0.415 

0.485 

0.555 

0.625 

0.695 

1.403 

2.132 

If  the  pressure  is  read  on  a  mercury  manometer  at  20°,  the 
heights  in  mercurial  centimeters  may  be  converted  into  grams 
per  square  centimeter  by  means  of  Table  III. 


300 


FLUIDITY  AND  PLASTICITY 


TABLE  III. — VALUES  OF  M.    PRESSURES  IN  GRAMS  PER  SQUARE  CENTI- 
METER, FOR  HEIGHTS  IN  MERCURIAL  CENTIMETERS 


Height,  centi- 
meters of 
mercury 

.0 

.1 

.2 

.3 

:• 

.5 

.6 

.7 

.8 

•• 

10 

135.4 

36.8 

38.2 

39.5 

40.9 

42.  2      43.  6 

44.9 

46.3 

47.6 

11 

49.0 

50.3 

51.7 

53.1 

54.4 

55.8 

57.1 

58.  5      59.  8 

61.2 

12 

62.5 

63.9 

65.2 

66.6 

68.0 

69.3 

70.7 

72.0  !  73.4 

74.7 

13 

76.1 

77.4 

78.8 

80.1 

81.5 

82.9 

84.2 

85.6  :   86.9 

88.3 

14 

89.6 

91'.  0 

92.3 

93.7 

95.0 

96.4 

97.8 

99.1   "00.5  "01.8 

15 

203.  2 

04.5 

05.9 

07.2 

08.6 

09.9 

11.3 

12.7 

14.0 

15.4 

16 

16.7 

18.1 

19.4 

20.8 

22.1 

23.5 

24.8 

26.2 

27.6 

28.9 

17 

30.3 

31.6 

33.0 

34.3 

35.7 

37.0 

38.4 

39.7 

41.  1 

42.5 

18 

43.8 

45.2 

46.5 

47.9 

49.2 

50.6 

51.9 

53.3 

54.6 

56.0 

19 

57.4 

58.7 

60.1 

61.4 

62.8 

64.1 

65.5 

66.8 

68.2 

69.5 

20 

70.9 

72.2 

73.6 

75.0 

76.3 

77.7 

79.0 

80.4 

81.7 

83.1 

21 

84.4 

85.8 

87.2 

88.5 

89.9 

91.2 

92.6 

93.9 

95.3 

96.6 

22 

98.0 

99.3 

*00.7 

•02.0 

*03.4 

•04.8 

*06.1 

*07.5 

•08.8 

*10.2 

23 

311.5 

12.9 

14.2 

15.6 

16.9 

18.3 

19.7 

21.0 

22.4 

23.7 

24 

25.1 

26.4 

27.8 

29.1 

30.5 

31.8 

33.2 

34.6 

35.9 

37.3 

25 

38.6 

40.0 

41.3 

42.7 

44.0 

45.4 

46.7 

48.  1 

49.5 

50.8 

26 

52.2 

53.5 

54.9 

56.2 

57.6 

58.9 

60.3 

61.6 

63.0 

64.4 

27 

65.7 

67.1 

68.4 

69.8 

71.1 

72.5 

73.8 

75.2 

76.5 

77.9 

28 

79.2 

80.6 

82.0 

83.3 

84.7 

86.0 

87.4 

88.7 

90.  1 

91.4 

29 

92.8 

94.2 

95.5 

96.9 

98.2 

99.6 

*00.9 

*02.3 

*03.  6 

*05.0 

30 

406.3 

07.7 

09.1 

10.4 

11.8 

13.1 

14.5 

15.8 

17.2 

18.5 

31 

19.9 

21.2 

22.6 

24.0 

25.3 

26.7 

28.0 

29.4 

30.7 

32.1 

32 

33.4 

34.8 

36.1 

37.5 

38.9 

40.2 

41.6 

42.9 

44.3 

45.6 

33 

47.0 

48.3 

49.7 

51.0 

52.4 

53.7 

55.1 

56.5 

57.8 

59.2 

34 

60.5 

61.9 

63.2 

64.6 

,  65.9 

67.3 

68.6 

70.0 

71.4 

72.  7 

35 

Oft 

74.1 

75.4 

76.8 

78.1 

79.5 

80.8 

82.2 

83.5 

84.9 

86.3 

nn    o 

1.4 

OO 

37 

87.  6 
501.2 

89.  0 
02.5 

90.  3 
03.9 

91  .  7 
05.2 

93.  0 
06.6 

94.  4 
07.9 

95.  7 
09.3 

97.  1 
10.6 

98.  4 
12.0 

99.  8 
13.3    ( 

.i'o.i 

38 

14.7 

16.1 

17.4 

18.8 

20.1 

21.5 

22.8 

24.2 

25.5 

26.9    ( 

.2  0.3 

39 

28.2 

29.6 

31.0 

32.3 

33.7 

35.0 

36.4 

37.7 

39.1 

40.4    ( 

.3  0.5 

( 

.4  0.6 

40 

41.8 

43.1 

44.5 

45.9 

47.2 

48.6 

49.9 

51.3 

52.6 

54.0    C 

.5  0.7 

41 

55.3 

56.7 

58.0 

59.4 

60.8 

62.1 

63.5 

64.8 

66.2 

67.5    ( 

.6  0.8 

42 

68.9 

70.2 

71.6 

72.9 

74.3 

75.6 

77.0 

78.4 

79.7 

81.1    ( 

.7  1.0 

43 

82.4 

83.8 

85.  1 

86.5 

87.8 

89.2 

90.5 

91.9 

93.3 

94.6    ( 

.8  1.1 

44 

96.0 

97.3 

98.7 

•oo.o 

*01.4 

*02.7 

*04.1 

*05.4 

*06.8 

*08.2    C 

.9  1.3 

45 

609.5 

10.9 

12.2 

13.6 

14.9 

16.3 

17.6 

19.0 

20.3 

21.7  L 

46 

23.1 

24.4 

25.8 

27.1 

28.5 

29.8 

31.2 

32.5 

33.9 

35.2 

47 

36.6 

38.0 

39.3 

40.7 

42.0 

43.4 

44.7 

46.1 

47.4 

48.8 

48 

50.1 

51.5 

52.9 

54.2 

55.6 

56.9 

58.3 

59.6 

61.0 

62.3 

49 

63.7 

65.0 

66.4 

67.8 

69.1 

70.5 

71.8 

73.2 

74.5 

75.9 

50 

77.2 

78.6 

79.9 

81.3 

82.6 

84.0 

85.4 

86.7 

88.1 

89.4 

51 

90.8 

92.1 

93.5 

94.8 

96.2 

97.5 

98.9 

•00.3 

•01.6 

*03.0 

52 

704.3 

05.7 

07.0 

08.4 

09.7 

11.1 

12.4 

13.8 

15.2 

10.5 

53 

17.9 

19.2 

20.6 

21.9 

23.3 

24.6 

26.0 

27.3 

28.7 

30.1 

54 

31.4 

32.8 

34.  1 

35.5 

36.8 

38.2 

40.9 

42.2 

43.6 

55 

45.0 

46.3 

47.7 

49.0 

50.4 

51.7 

53!  1 

54.4 

55.8 

57.1 

56 

58.5 

59.9 

61.2 

62.6 

63.9 

65.3 

66.6 

68.0 

69.3 

70.7 

57 

72.0 

73.4 

74.7 

76.  1 

77.5 

78.8 

80.2 

81.5 

82.9 

84.2 

58 

85.6 

86.9 

88.3 

89.6 

91.0 

92.4 

93.7 

95.1 

96.4 

97.8 

59 

99.1 

•00.5 

•01.8 

*03.2 

*04.5 

*05.9 

•07.3 

•08.6 

*10.0 

•11.3 

60 

812.7 

14.0 

15.4 

16.7 

18.1 

19.4 

20.8 

22.2 

23.5 

24.9 

61 

26.2 

27.6 

28.9 

30.3 

31.6 

33.0 

34.3 

35.7 

37.1 

38.4 

62 

39.8 

41.  1 

42.5 

43.8 

45.2 

46.5 

47.9 

49.2 

50.6 

52.0 

63 

53.3 

54.7 

56.0 

57.4 

58.7 

60.1 

61.4 

62.8 

64.  1 

65.5 

64 

66.8 

68.2 

70.9 

72.3 

73.6 

75.0 

76.3 

77.7 

79.0 

65 

80.4 

81.7 

83!  1 

84.5 

85.8 

87.2 

88.5 

89.9 

91.2 

92.6 

66 

93.9 

95.3 

96.6 

98.0 

99.4 

*00.7 

*02.1 

•03.4 

•04.8 

•06.1 

67 

907.5 

08.8 

10.2 

11.5 

12.9 

14.3 

15.6 

17.0 

18.3 

19.7 

68 

21.0 

22.4 

23.7 

25.1 

26.4 

27.8 

29.2 

30.5 

31.9 

33.2 

69 

34.6 

35.9 

37.3 

38.6 

40.0 

41.3 

42.7 

44.  1 

45.4 

46.8 

APPENDIX  A 


301 


TABLE  III. — Continued 


Height,  centi- 
meters of 
mercury 

.0 

.  1 

1 
.2           .3           .4           .5     |      .6 

1              1              1 

.7 

.8 

.9 

70 

48.1 

49.5      50.8 

52.2 

53.5 

54.9 

56.2 

57.6 

58.9 

60.3 

71 

61.7 

63.0  I   64.4 

65.7 

67.1 

68.4 

69.8 

71.1 

72.5 

73.8 

72 

75.2 

76.6      77.9 

79.3 

80.6 

82.0 

83.3 

84.7 

86.0 

87.4 

73 

88.7 

90.1 

91.5 

92.8 

94.2 

95.5 

96.9 

98.2 

99.6 

•00.9 

74 

1002.3 

03.6 

05.0 

06.4 

07.7 

09.1 

10.4 

11.8 

13.1 

14.5 

75 

15.8 

17.2 

18.5 

19.9 

21.2 

22.6 

24.0 

25.3 

26.7 

28.0 

76 

29.4 

30.7 

32.1 

33.4 

34.8 

36.1 

37.5 

38.9 

40.2 

41.6 

77 

42.9 

44.3 

45.6 

47.0 

48.3 

49.7 

51.0 

52.4 

53.8 

55.1 

78 

56.5 

57.8 

59.2 

60.5 

61.9 

63.2 

64.6 

65.9 

67.3 

68.7 

79 

70.0 

71.4 

72.7 

74.1 

75.4 

76.8 

78.1 

79.5 

80.8 

82.2 

80 

83.6 

84.9 

86.3 

87.6 

89.0 

90.3 

91.7 

93.0 

94.4 

95.7 

81 

97.  li  98.4 

99.8 

•01.2 

•02.5 

•03.9 

'05.2 

*06.6 

•07.9 

*09.3 

82 
83 

1110.  6|    12.0 
24.21   25.5 

13.  3 
26.9 

14.  7 
28.2 

16.  1 
29.6 

17.  4 
31.0 

18.  8 
32.3 

20.  1 
33.7 

21.  5 
35.0 

22.  8 
36.4 

1.4 

84 

37.7!   39.1 

40.4 

41.8 

43.1 

44.5 

45.9 

47.2 

48.6 

49.9 

85 

51.3    52.6 

54.0 

55.3 

56.7 

58.0 

59.4 

60.7 

62.1 

63.5 

0.  1 

0.  1 

86 

64.  8i   66.2 

67.5 

68.9 

70.2 

71.6 

72.9 

74.3 

75.6 

77.0 

D.2 

0.3 

87 

78.  4i   79.7 

81.1 

82.4 

83.8 

85.1 

86.5 

87.8 

89.2 

90.5 

5.8 

0^5 

88 

91.9 

93.3 

94.6 

96.0 

97.3 

98.7 

•oo.o 

*01.4 

•02.7 

*04.1 

04 

06 

89 

1205.4 

06.8 

08.2 

09.5 

10.9 

12.2 

13.6 

14.9 

16.3 

17.6 

0.5 

0.7 

90 

19.0 

20.3 

21.7 

23.1 

24.4 

25.8 

27.1 

28.5 

29.8 

31.2 

0.6 
0.7 

0.8 
1.0 

91 

32.5 

33.9 

35.2 

36.6 

37.9 

39.3 

40.7 

42.0 

43.4 

44.7 

0  8 

1  1 

92 

46.1 

47.4 

48.8 

50.1 

51.5 

52.8 

54.2 

55.6 

56.9 

58.3 

09 

13 

93 

59.6 

61.0 

62.3 

63.7 

65.0 

66.4 

67.7 

69.1 

70.5 

71.8 

94 

73.2 

74.5 

75.9 

77.2 

78.6 

79.9 

81.3 

82.6 

84.0 

85.4 

95 

86.7 

88.1 

89.4 

90.8 

92.1 

93.5 

94.8 

96.2 

97.5 

98.9 

96 

1300.2 

01.6 

03.0 

04.3 

05.7 

07.0 

08.4 

09.7 

11.1 

12.4 

97 

13.8 

15.1 

16.5 

17.9 

19.2 

20.6 

21.9 

23.3 

24.6 

26.0 

98 

27.3 

28.7 

30.0 

31.4 

32.8 

34.1 

35.5 

36.8 

38.2 

39.5 

99 

40.9 

42.2 

43.6 

44.9 

46.3 

47.7 

49.0 

50.4 

51.7 

53.1 

100 

54.4 

55.8 

57.1 

58.5 

59.8 

61.2 

62.5 

63.9 

65.3 

66.6 

200 

2708.  7 

10.0 

11.4 

12.7 

14.1 

15.5 

16.8 

18.2 

19.5 

20.9 

300 

4062.8 

64.2 

65.5 

66.9 

68.2 

69.6 

71.0 

72.3 

73.7 

75.0 

302 


FLUIDITY  AND  PLASTICITY 


If  the  temperature  of  the  mercury  is  other  than  20°  a  correc- 
tion is  applied  using  Table  IV. 


TABLE   IV. — VALUES  OF   N.     CORRECTION   IN   PRESSURES   (GRAMS   PEI 

SQUARE  CENTIMETER)  FOR  VARIOUS  TEMPERATURES  AND  MERCURIAL 

HEIGHTS 


Temperature, 
degrees 
Centigrade 

Height  of  mercury,  centimeters 

10 

20 

30 

40 

50 

60 

70 

80 

90 

100 

5 

0.4 

0.7 

1.1 

.5 

.8 

2.2 

2.6 

2.9 

3.3 

3.7 

6 

0.3 

0.7 

1.0 

.4 

.7 

2.1 

2.4 

2.7 

3.1 

3.4 

7 

0.3 

0.6 

1.0 

.3 

.6 

.9 

2.2 

2.6 

2.9 

3.2 

8 

0.3 

0.6 

0.9 

.2 

.5 

.8 

2.1 

2.4 

2.6 

2.9 

9 

0.3 

0.5 

0.8 

.1 

.4 

.6 

1.9 

2.2 

2.4 

2.7 

10 

0.2 

0.5 

0.7 

1.0 

.2 

.5 

1.7 

2.0 

2.2 

2.4 

11 

0.2 

0.4 

0.7 

0.9 

.1 

.3 

1.5 

1.8 

2.0 

2.2 

12 

0.2 

0.4 

0.6 

0.8 

.0 

.2 

1.4 

1.6 

1.8 

2.0 

13 

0.2 

0.3 

0.5 

0.7 

0.9 

1.0 

1.2 

1.4 

1.5 

1.7 

14 

0.2 

0.3 

0.4 

0.6 

0.7 

0.9 

1.0 

1.2 

1.3 

1.5 

15 

0.1 

0.2 

0.4 

0.5 

0.6 

0.7 

0.9 

1.0 

1.1 

1.2 

16 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

1.0 

17 

0.1 

0.1 

0.2 

0.3 

0.4 

0.4 

0.5 

0.6 

0.7 

0.7 

.       18 

0.1 

0.1 

0.2 

0.2 

0.2 

0.3 

0.3 

0.4 

0.4 

0.5 

19 

0.0 

0.0 

0.1 

0.1 

0.1 

0.2 

0.2 

0.2 

0.2 

0.2 

20 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

21 

0.0 

0.0 

-0.1 

-0.1 

-0.1 

-0.2 

-0.2 

-0.2 

-0.2 

-0.2 

22 

-0.1 

-0.1 

-0.1 

-0.2 

-0.2 

-0.3 

-0.3 

-0.4 

-0.4 

-0.5 

23 

-0.1 

-0.1 

-0.2 

-0.3 

-0.4 

-0.4 

-0.5 

-0.6 

-0.7 

-0.7 

24 

-0.1 

-0.2 

-0.3 

-0.4 

-0.5 

-0.6 

-0.7 

-0.8 

-0.9 

-1.0 

25 

-0.1 

-0.2 

-0.4 

-0.5 

-0.6 

-0.7 

-0.9 

-    .0 

-1.1 

-1.2 

26 

-0.2 

-0.3 

-0.4 

-0.6 

-0.7 

-0.9 

-    .0 

-    .2 

-1.3 

-1.5 

27 

-0.2 

-0.3 

-0.5 

-0.7 

-0.9 

-    .0 

-    .2 

-    .4 

-1.5 

-1.7 

28 

-0.2 

-0.4 

-0.6 

-0.8 

-    .0 

-    .2 

-    .4 

-    .6 

-1.8 

-2.0 

29 

-0.2 

-0.4 

-0.7 

-0.9 

-    .1 

-    .3 

-    .5 

-    .8 

-2.0 

-2.2 

30 

-0.2 

-0.5 

-0.7 

-    .0 

-    .2 

-    .5 

-   .7 

-2.0 

-2.2 

-2.4 

31 

-0.3 

-0.5 

-0.8 

-    .1 

-    .4 

-    .6 

-    .9 

-2.2 

-2.4 

-2.7 

32 

-0.3 

-0.6 

-0.9 

-    .2 

-    .5 

-    .8 

-2.1 

-2.4 

-2.6 

-2.9 

33 

-0.3 

-0.6 

-1.0 

-    .3 

-    .6 

-1.9 

-2.2 

-2.6 

-2.9 

-3.2 

34 

-0.3 

-0.7 

-1.0 

-    .4 

-    .7 

-2.1 

-2.4 

-2.7 

-3.1 

-3.4 

35 

-0.4 

-0.7 

-1.1 

-    .5 

-   .8 

-2.2 

-2.6 

-2.9 

-3.3 

-3.7 

APPENDIX  A  303 

The  correction  for  the  difference  in  level  between  the  middle 
of  the  manometer  and  the  viscometer  is  made  negligible  in  setting 
up  the  apparatus. 

MEASUREMENT  OF  TIME 

We  have  seen  that  the  pressure  in  grams  per  square  centimeter 
must  always  be  30  times  as  great  as  the  distance  between  the 
bulbs.  On  the  other  hand  the  pressure  must  always  be  kept  small 
enough  so  that  the  time  of  flow  can  be  measured  to  the  desired 
accuracy.  Thus  the  time  should  not  fall  below  200  sec.  since  one 
cannot  measure  the  time  more  accurately  than  to  0.2  sec.  with  a 
stop-watch. 

The  stop-watch  should  be  tested  repeatedly  against  the  second 
hand  of  a  good  time  piece.  It  should  not  gain  or  lose  as  much  as 
0.2  sec.  in  5  min.  It  is  well  to  keep  the  watch  in  the  same  posi- 
tion during  successive  measurements,  as  well  as  not  to  allow  it  to 
be  nearly  run  down  during  a  measurement.  In  selecting  a  stop- 
watch it  should  be  noted  that  watches  show  better  performance 
whose  mechanism  continues  to  run  whether  the  split-second  hand 
is  in  use  or  not.  The  performance  of  the  watch  may  be  tested 
at  the  U.  S.  Bureau  of  Standards. 

TEMPERATURE  • 

The  viscometer  is  kept  at  a  constant  temperature  by  means  of  a 
large,  well-stirred  bath  which  is  regulated  by  hand,  if  a  series  of 
temperatures  are  to  be  measured,  or  by  a  thermostat,  if  the  bath 
is  to  be  used  for  a  long  time  at  a  single  temperature.  Since  at 
0°  the  fluidity  of  water  increases  0.1  per  cent  for  every  0.03° 
rise  in  temperature  it  is  clear  that  the  temperature  regulation 
must  be  to  at  least  0.03°.  For  more  viscous  substances  a  still 
more  precise  regulation  is  necessary  if  the  same  degree  of  accu- 
racy is  to  be  obtained. 

A  thermometer  should  be  used  which  is  graduaded  to  tenths 
and  calibrated  through  its  entire  length.  The  ice  point  should 
be  determined  from  time  to  time.  If  it  is  impracticable  to  have 
the  entire  thread  of  mercury  immersed  at  all  times  a  correction 
should  be  made  for  the  emergent  stem.  The  following  table 
may  be  used: 


304 


.FLUIDITY  AND  PLASTICITY 


TABLE  V. — CORRECTION  OF  A  NORMAL  THERMOMETER  FROM  0°  TO  100°C 
FOR  EMERGENT  STEAM  GRADUATED  IN  TENTHS  OF  A  DEGREE 


Difference  in  temperature  between  mean  temperature 

Number  of  de- 

of emergent  steam  and  bath.     Corrections  in  degrees  to  be 

grees  of  mercury 

added  to  the  observed  temperature 

30° 

40° 

50° 

60° 

70° 

80° 

10 

0.05 

0.05 

0.05 

0.05 

0.10 

0.10 

20 

0.10 

0.15 

0.15 

0.15 

0.20 

0.20 

30 

0.20 

0.25 

0.25 

0.25 

0.30 

0.35 

40 

0.30 

0.30 

0.35 

0.40 

0.45 

0.50 

50 

0.35 

0.40 

0.45 

0.50 

0.55 

0.60 

Since  measurements  are  always  preferred  for  even  degrees  it  is 
a  great  advantage  for  the  worker  to  have  on  the  bath  before  him 
a  table  showing  what  temperatures  on  the  thermometer  must  be 
employed  in  order  to  obtain  a  desired  even  temperature.  The 
temperatures  of  0°,  10°,  20°,  40°,  60°,  80°,  100°  are  sufficient  to 
give  a  good  curve  over  this  range. 

THE  PRESSURE  REGULATOR 

Viscosity  measurements  have  usually  been  carried  out  without 
the  use  of  a  pressure  regulator,  but  due  to  the  withdrawal  of  the 
air  in  use  and  to  possible  small  leaks  in  the  connections  and  to 
changes  in  temperature,  the  pressure  rises  and  falls  and  is  hardly 
ever  constant  during  the  time  of  a  single  measurement.  With 
a  pressure  regulator  the  pressure  will  often  stay  constant  to  the 
limit  of  the  experimental  error  for  a  day  or  more  at  a  time,  with- 
out temperature  regulation  of  the  room,  heat  insulation  of  the 
apparatus  or  any  particular  care  in  using  the  air.  Not  only  is 
this  a  saving  of  time  and  annoyance  to  the  experimenter  but  by 
using  only  a  few  pressures  at  the  most  there  is  a  considerable 
saving  of  time  in  calculation.  Hence  the  pressure  regulator 
is  a  necessity  for  extended  work. 

The  diagrammatic  view  of  the  apparatus  with  pressure  regula- 
tor is  given  in  Fig.  92.  Air  is  forced  in  through  a  needle  valve  A 
to  a  storage  reservoir  B  whose  pressure  in  pounds  per  square  inch 
is  shown  on  the  gauge  C.  In  adjusting  the  pressure  regulator 
the  air  is  very  slowly  admitted  to  the  stabilizing  reservoir  F  by 


APPENDIX  A 


305 


means  of  the  needle  valve  D.  The  valve  E  is  convenient  in  locat- 
ing leaks  in  the  apparatus,  etc.,  but  is  not  often  used.  The  valve 
G  is  a  direct  connection  to  air  which  is  also  seldom  used. 

The  pressure  regulator  consists  of  five  brass  tubes  6  cm.  in 
diameter  which  are  filled  with  water  let  in  at  K,  the  valves  Of, 
0"  etc.  being  open  and  the  valve  N  closed.  When  the  water 
begins  to  overflow  at  M  into  the  drain  pipe,  the  water  is  shut  off 


FIG.  92. — Diagram  of  viscometer  set-up  with  multiple   tube  water  stabilizer. 

at  K,  and  as  soon  as  equilibrium  is  reached,  the  drain  pipe  is  also 
closed  off  at  Z  and  the  valves  0',  0",  etc.  are  closed. 

By  allowing  air  to  pass  very  slowly  through  the  valve  D  the  air 
will  be  gradually  forced  down  the  tube  H'  until  it  bubbles  out 
through  the  water,  and,  if  the  pet-cock  «/'  is  open,  into  the  air. 
If  the  stream  of  air  is  very  slow,  say  a  bubble  or  two  per  second, 
it  is  evident  that  the  pressure  will  be  constant.  If  a  higher  pres- 
sure is  desired  the  pet-cock «/'  is  closed  when  the  pressure  becomes 
the  sum  of  the  pressures  obtained  by  the  two  tubes  separately 
and  so  on  for  the  five  different  pressures  up  to  the  maximum 
capacity  of  the  regulator.  In  lowering  the  pressure  one  must 
be  careful  to  turn  the  pet-cocks  to  air  in  the  reverse  order  «7V 
«7IV  J111  and  Ju  Jl  in  order  that  the  air  under  pressure  may  not 
cause  the  water  to  be  drawn  back  into  the  system.  The  advan- 

20 


306  FLUIDITY  AND  PLASTICITY 

tage  of  the  drain  pipe  U  is  that  of  securing  day  by  day  practically 
identical  pressures,  without  the  loss  of  time  in  adjustment.  If 
other  pressures  than  these  are  desired,  they  may  be  obtained  by 
drawing  off  some  of  the  water  from  one  or  more  of  the  stand  pipes. 
The  glass  gage  at «/',  etc.,  aid  the  manipulator  in  adjusting  the  cur- 
rent of  air.  They  may  be  cleaned  by  unscrewing  the  pet-cocks 
above  and  using  a  small  brush. 

The  beginner  must  be  cautioned  particularly  against  turning 
the  system  to  air  at  the  viscometer  since  it  may  result  in  filling 
the  manometer,  etc.  with  water.  To  prevent  such  an  accident 
and  to  dry  the  air,  the  reservoir  P  containing  granular  calcium 
chloride  is  introduced.  Any  liquid  should  be  drained  at  intervals. 

THE  MANOMETER 

The  manometer  consists  of  a  plate  glass  mirror  which  must  be 
mounted  vertically,  on  which  is  stretched  a  2-m  steel  tape 
graduated  in  millimeters.  Over  the  tape  is  fixed  the  glass  tube 
of  the  manometer  bent  so  that  both  the  right  and  left  limbs  may 
be  read  on  the  same  tape.  The  manometer  may  be  filled  with 
either  mercury  or  water.  If  water  is  used  for  low  pressures 
another  manometer  will  be  desired  for  mercury.  Since  it  is 
possible  to  read  the  manometer  to  0.01  cm  one  can  use  the  mer- 
cury manometer  down  to  10  cm  (135  g  per  square  centimeter) 
with  the  desired  accuracy.  With  water  one  can  go  down  to 
about  50  g  per  square  centimeter,  but  not  much  further  unless  a 
correction  is  made  for  the  true  average  pressure.  A  thermometer 
near  the  middle  of  the  manometer  is  needed  to  give  the  tem- 
perature of  the  manometer  fluid. 

THE  BATH 

The  viscometer  V  is  mounted  on  a  massive  brass  frame  Fig. 
93  by  means  of  brass  clips  designed  especially  for  this  purpose. 
The  frame  slides  in  grooves  on  the  side  of  the  bath  so  that  the 
viscometer  may  be  easily  kept  in  a  vertical  position.  The  viscom- 
eter is  connected  by  heavy-walled  rubber  tubing  to  the  pressure 
by  way  of  the  three-way  glass  stop  cocks  L  and  R,  the  third 
connection  being  to  air.  The  temperature  of  the  bath  is  raised 
by  means  of  a  burner  W  which  is  connected  without  the  use  of 
rubber  to  the  gas  supply.  The  second  burner  Y  with  stop  cock 
and  pilot  flame  is  used  as  needed  to  obtain  the  fine  regulation. 


APPENDIX  A 


307 


<?'.  EM  ? 


FIG.  93. — Details  of  hath,  frame,  and  clips  for  holding  viscometer. 


308 


FLUIDITY  AND  PLASTICITY 


To  assist  in  the  regulation,  cold  water  is  admitted,  when  desired, 

by  a  cock  at  S.     A  drain  pipe,  Q,  maintains  the  bath  at  a  constant 

level.     It  may  also  be  unscrewed  to  permit  draining  the  water 

from  the  bath.     The  bath  is  insulated  on  two  sides. 

THE  DENSITY 

It  is  not  necessary  to  know  the  exact  density  in 
order  to  obtain  the  fluidity  by  this  method.  But 
the  density  can  be  measured  at  the  same  time 
with  accuracy  with  little  additional  labor.  Since 
the  fluidity  is  very  closely  related  to  the  volume, 
according  to  the  law  that  the  fluidity  is  directly 
proportional  to  the  free  volume,  the  specific  volume 
should  usually  be  obtained  with  precision. 

The  instrument  shown  in  Fig.  94  is  convenient  to 
use  and  unlike  the  Sprengel  pycnometer,  it  can  be 
used  to  determine  the  density  below  room  tem- 
perature. It  is  filled  to  the  mark  with  water  and 
weighed  at  every  temperature  at  which  it  is  to 
be  used.  It  is  then  cleaned,  dried,  weighed,  and 
filled  with  the  liquid  to  be  determined  and  again 
weighed.  The  ratio  of  the  weights  of  liquid  cor- 
FIG.  94.—  rected  simply  for  the  buoyancy  of  the  air  gives  the 

A  pycnometer    correct  specific  gravity  referred  to  water  at  4°C. 
The  densities  of  water  are  given  in  Table  VI. 

TABLE  VI. — DENSITY  AND  VOLUME  OF  WATER  IN  GRAMS  PER  MILLILITER 


Temperature 

Density 

Logarithm 
density 

Specific  volume 

0 

0.99987 

9.99994-10 

1.00013 

10 

0.99973 

9.99988-10 

1.00027 

20 

0.99823 

9.99923-10 

1.00177 

30 

0.99568 

9.99811-10 

1  .  00435 

40 

0.99225 

9.99662-10 

1  .  00782 

50 

0.98807 

9.99479-10 

.  01207 

60 

0.98324 

9.99266-10 

.01705 

70 

0.97781 

9.99025-10 

.02270 

80 

0.97183 

9.98759-10 

.02899 

90 

0.96534 

9.98468-10 

.03590 

100 

0.95838 

9.98154-10 

.04343 

APPENDIX  A 


309 


The  formula  to  be  used  in  obtaining  the  density  is: 


=  —  po  +0.0012 


/W     —  W0\ 
\          Wo       / 


where  w'  =  weight  of  liquid  at  t°C, 
w0  =  weight  of  water  at  f°C, 
PO  =  density  of  water  at  t°C. 
The  liquid  is  introduced  or  removed  from  the  pycnometer  by 


TO -SUCTION 


FIG.  95. — Apparatus  for  cleaning  and  filling  viscometer. 


means  of  the  capillary  pipette,  used  also  for  introducing  liquid 
into  the  viscometer,  shown  in  Fig.  95. 

This  rubber  tubing  as  well  as  the  heavy  walled  tubing  at  the 
top  of  the  viscometer  should  be  scrupulously  cleaned  on  the 
inside  to  remove  dust  before  they  are  used. 

If  the  capillary  stem  of  a  25  ml  pycnometer  has  a  bore  of  0.08 
cm  it  is  capable  of  an  accuracy  of  0.01  per  cent  by  reading  the 
meniscus  to  within  1  mm.  It  is  well  to  have  two  pycnometers 


310 


FLUIDITY  AND  PLASTICITY 


of  equal  size  and  employ  the  tare  method  in  weighing. 

Strictly,  it  is  necessary  to  measure  the  density  at  only  one 
temperature  by  this  method.  The  working  volume  of  the  vis- 
cometer  has  to  be  adjusted  each  time  that  the  temperature  of  the 
liquid  is  raised.  By  noting  the  expansion  of  this  working  volume 
for  each  temperature  interval  it  is  readily  possible  to  calculate 
the  specific  volume  and  density.  The  portion  of  the  viscometer 
HG,  Fig.  29,  is  graduated  in  millimeters.  By  filling  the  viscometer 
with  mercury  from  A  to  G,  and  weighing  this  mercury,  the  work- 
ing volume  V  can  be  actually  determined.  And  by  filling  a 
given  length  of  the  capillary  HG  with  mercury,  the  volume  v'  of 
the  capillary  per  centimeter  is  easily  determined.  The  density 
of  mercury  is  given  in  Table  III. 

TABLE    VII. — DENSITY    AND    VOLUME    OF    MERCURY    IN    GRAMS    PER 

MlLLILITER 


Temperature, 
degrees 

Density 

Logarithm 
density 

Specific  volume 

10 

13.570 

.  13260 

0.073687 

15 

13.558 

.  13220 

0.073757 

20 

13.546 

.13181 

0.073822 

25 

13.534 

.13142 

0.073887 

30 

13.522 

.  13104 

0.073954 

If,  therefore,  the  specific  volume  of  the  liquid  is  s0  at  temperature 
t0  and  on  forcing  the  meniscus  at  the  left  just  up  to  the  trap,  the 
right  meniscus  is  a  distance  d  away  from  its  proper  level  G,  then 
at  the  new  temperature  t,  the  specific  volume  s  must  be 

.--Jj(i+^)-J  do) 

With  this  volumeter  it  must  be  remembered  that  the  errors 
are  cumulative.  On  the  other  hand  with  the  pycnometer  method 
care  must  be  taken  to  wipe  off  drops  of  liquid  which  may  adhere 
to  the  inside  of  the  glass,  and  to  prevent  the  evaporation  of 
volatile  substances,  on  account  of  which  a  stopper  is  added  to  the 
pycnometer. 

Assuming  that  a  capillary  is  used  whose  radius  is  0.01  cm  and 
that  the  tube  HG  has  a  radius  which  is  ten-fold  this  amount,  or 


APPENDIX  A  311 

0.1  cm  (cf.  page  319)  reading  the  meniscus  to  0.1  mm  will  give 
an  accuracy  in  the  specific  volume  of  0.01  per  cent. 

CLEANING  AND  FILLING  THE  VISCOMETER 

The  viscometer  is  not  removed  from  its  frame  during  the  course 
of  an  investigation.  Two  hooks  are  screwed  into  a  board  on  the 
wall  which  will  hold  the  viscometer  frame  firmly  at  E,  Fig.  95. 
Chromic  acid,  added  with  pipette,  is  drawn  through  the  instru- 
ment by  means  of  suction.  The  frame  and  viscometer  are  then 
again  placed  on  hooks  in  an  inverted  position  D  and  the  liquid 
withdrawn  by  means  of  suction.  A  Woulff  flask  is  interposed 
between  the  rubber  tubing  and  the  suction  line.  The  apparatus 
is  washed  out  repeatedly  with  dust-free  water  and  finally  with 
dust-free  alcohol  and  dust-  and  grease-free  ether.  Air  which  has 
passed  over  granulated  calcium  chloride  A  and  through  a  long 
column  of  absorbent  cotton  B  is  then  drawn  through  using  clean 
rubber  tubing. 

To  fill  the  instrument  an  amount  of  liquid  slightly  greater  than 
the  working  volume  is  drawn  up  into  the  clean  pipette  F  which  is 
wiped  free  of  dust  by  means  of  chamois  skin  just  before  use.  The 
liquid  is  protected  from  the  moisture  of  the  air  by  means  of  the 
drying  tube  containing  calcium  chloride  held  in  position  by  means 
of  absorbent  cotton. 

THE  VISCOSITY  RECORD 

The  data  may  be  kept  on  sheets  ruled  somewhat  as  follows : 
which  will  give  a  compact  and  systematic  record  of  both  data 
and  the  calculations: 


312 


FLUIDITY  AND  PLASTICITY 


TABLE    VIII. — LAFAYETTE     COLLEGE     VISCOSITY    RECORD 
Page  I 

Substance       Pure  Water  Remarks        Calibration 

Date Observer       W.  G.  K.  =  .0 .46 

Viscometer  No.  J>_,     Pycnometer  No.    2 

Log  C'  =  8  37698-10  Log  ~  =  6  22122 


Time 

Manometer, 

upper  read- 

Temperature 
bath 

Limb 

Min- 

Sec- 

Time, 
sec- 
onds 

ing,    lower 
reading 

Temper- 
ature 

differ- 
ence = 

Weight 
pyc. 

utes 

onds 

ho 

Start 

Finish 

20 

L 

5 

7.0 

307.0 

259.46 

259.46 

21.2 

287  .  58 

28.12 

231.34 

20 

R 

5 

8.2 

308.2 

259.48 

259.48 

21.1 

287.60 

28.12 

231.36 

' 

hip 

K 

±hi 
PK±L 

Po 

cv 

Ct» 

P 

1 
in  cp 

<f 

• 

Remarks 

0.9982 

+  .44 

-.79 

-.35 

230.99 

,„ 

229.23 

1.0050 

99.50 

water  mano- 

-.44 

-.79 

-1.23 

230.13 

1.75 

228.38 

1.0052 

99.48 

meter 

CALCULATION  OF  CONSTANTS 

Let  us  use  the  above  data  for  water  at  20°  to  show  the  method 
of  calculation  of  constants,  etc.  We  record  the  sum  of  the  upper 
and  lower  manometer  readings  merely  as  a  check  against  error 
in  reading,  since  this  sum  should  be  constant.  With  our  instru- 
ment V  =  4.0  ml,  and  I  =  7.5  cm  hence  C'  =  0.02377.  The 
value  of  231.34,  corrected  by  Table  II  for  K  gives 

200  cm  at  21.2°  =  0.69 

30  cm  at  21.2°  =  0.10 

1.34  cm  at  21.2°  =  0.00 

Total  correction  =  0.79 

pl  =  231.34  -  0.79  =  230.55  cm 


APPENDIX  A  313 

The  value  of  L  is  negligible.     Calculating  the  approximate  value 
of  C  using  Eq.  (3)  we  have, 

0.01005  X  307  +0.02377  X  0.998  _ 
230.55  X  307  X  307 

Calculating  now  the  hydrostatic  head,  using  this  value  of  C,  we 
have  from  Eq.  (5) 

hi  =  0.44  +  0.01  =  0.45. 

Now  p  =  230.55  +  0.45  =  231.0  for  the  left  limb  or 
=  230.57  -  0.45  =  230.1  for  the  right  limb; 

hence,  on  applying  again  Eq.  (3)  the  true  value  of  C  becomes 
c  _  0.1005  X307+  0.02377X0.998  _ 
231  X  307  X  307~ 

EXAMPLE  OF  CALCULATION  OF  VISCOSITY  AND  FLUIDITY 

Suppose  that  we  assume  that  we  had  given  the  constants  of 
the  apparatus,  and  that  we  desired  to  calculate  out  the  viscosity. 
We  have  hi  =  +  0.45,  K  =  —0.79,  so  that  the  corrected  pressure 
is  231.0.  We  may  now  apply  Eq.  (1)  at  once,  but  advantages 
may  be  obtained,  without  extra  labor,  by  calculating  the  value 
of  P  in  the  equation 

Cpt  -  C'p/t  =  CPT 

P  =  P-^  (ID 

which  is  evidently  the  pressure  consumed  in  overcoming  viscous 

f1 
resistance   solely.     In   this   case   -^  =  1.76   hence  P  =229.2. 

The  fluidity  <p  is 


Instead  of  writing  the  viscosity  as  0.01005  we  prefer  to  multiply 
by  100  and  record  the  datum  as  1.005  centipoises  (cp),  which  keeps 
most  viscosities  from  becoming  inconveniently  small  fractions 
and  it  also  makes  the  viscosities  "specific,"  referred  to  water  at 
practically  20°. 

In  scrutinizing  the  data  heretofore  published  on  viscosity  one 
is  particularly  interested  in  the  magnitude  of  the  kinetic  energy 
correction  and  it  may  be  subject  to  slight  changes  in  the  future. 


314  FLUIDITY  AND  PLASTICITY 

Publication  of  the  temperature,  time,  pressures  p  and  P,  density, 
viscosity,  and  fluidity  makes  the  data  quite  complete  and  cor- 
rection easy. 

In  constructing  the  viscometer,  the  glass  blower  must  select 
a  piece  of  capillary  tubing  which  has  not  only  a  uniform  bore 
but  also  one  which  has  a  radius  which  must  be  selected  within 
rather  narrow  limits.  This  requires  the  measurement  of  the 
radius,  which  is  accomplished  as  follows.  The  capillary  is  filled 
with  mercury  completely  to  a  distance  of  exactly  10  cm,  this 
mercury  is  then  run  out  on  to  a  watch  crystal  and  weighed.  The 
radius  of  the  capillary  in  centimeters  can  be  read  at  once  from 
Table  IX.  These  measurements  need  not  be  exact,  but  where  it 
is  desired  to  measure  the  average  radius  with  exactitude,  as  in 
absolute  measurement,  it  is  to  be  noted  that  the  volume  of  the 
mercury  is  calculated  for  20°C  and  that  the  values  are  corrected 
for  buoyancy  of  the  air  so  that  there  is  no  correction  in  weighing 
with  platinum  weights.  It  is  assumed  that  the  mercury  thread 
is  a  true  cylinder. 

Having  found  the  radius  of  the  capillary,  it  becomes  feasible 
to  cut  off  a  length  which  will  give  a  time  of  flow  of  not  less  than 
200  sec.  for  the  assumed  maximum  fluidity,  e.g.,  500,  with  a 
pressure  of  50  g  per  square  centimeter  and  a  volume  of  flow  of 
4  ml.  The  lengths  to  be  cut  off  for  capillaries  of  different  radii 
are  given  in  Table  X  (cf.  also  Fig.  24). 

The  table  shows  that  with  a  maximum  fluidity  of  500  and  a 
permissible  length  of  capillary  up  to  20  cm,  the  radius  must  not 
be  as  great  as  0.015  cm;  and  if  the  ratio  of  the  length  to  the  radius 
is  to  be  greater  than  500  in  order  to  minimize  "end  effects,"  the 
radius  must  be  over  0.010  cm,  which  limits  the  selection  within 
quite  narrow  limits. 

The  viscometer  with  a  500  capillary  will  serve  for  quite  viscous 
liquids,  for  the  pressure  can  be  varied  from  50  to  10,000  and  the 
time  may  conveniently  be  increased  fivefold,  hence  one  can 
measure  two-thousand-fold,  i.e.,  from  500  to  0.5.  Nevertheless 
in  an  investigation  in  which  no  fluidities  are  to  be  measured  above 
50,  it  is  convenient  to  use  a  viscometer  with  a  maximum  of  50, 
and  therefore  the  length  of  capillary  will  be  one-tenth  of  that 
indicated  by  Table  X.  Just  what  maximum  to  specify,  as  5,000 
500,  50,  5,  or  0.5,  may  easily  be  judged  by  the  use  of  Tables  XI 


APPENDIX  A 


315 


TABLE  IX. — THE  AVERAGE  RADIUS  OP  A  CAPILLARY  TUBE  IN  CENTIMETERS 

CORRESPONDING  TO  THE  WEIGHT  OF  MERCURY  REQUIRED  TO  FILL  A 

LENGTH    OP    10    CM    AT    20° 


Radius,  centi- 
meters 

Weight, 
grams 

Difference 

Radius, 
centimeters 

Weight, 
grams 

Difference 

0.001 

0.0004 

0.051 

.1068 

0.002 

0.0016 

12 

0.052 

.1506 

438 

0.003 

0.0038 

™ 

0.053 

.1953 

447 

0.004 
0.005 

0.0068 
0.0106 

oO 

38 

0.054 
0.055 

.2408 
.2872 

455 
464 

0.006 

0.0153 

CO 

0.056 

.3344 

472 

0.007 

0.0209 

ob 

0.057 

.3825 

481 

0.008 

0.0272 

63 
73 

0.058 

.4315 

490 

0.009 

0.0345 

0.059 

.4813 

498 

0.010 

0.0426 

81 
89 

0.060 

.5319 

50  G 
515 

0.011 

0.0515 

0.061 

.5834 

0.012 
0.013 

0.0613 
0.0719 

98 
106 

115 

0.062 
0.063 

.6357 
.6889 

523 
532 
541 

0.014 
0.015 
0.016 
0.017 
0.018 

0  .  0834 
0.0957 
0.1089 
0.1230 
0.1379 

123 
132 
141 
149 

0.064 
0.065 
0.066 
0.067 
0.068 

.7430 
.7979 
.8536 
.9102 
.9677 

549 
557 
566 
575 

0.019 

0.1536 

157 

0.069 

2.0259 

582 

0.020 

0.1702 

166 

175 

0.070 

2.0851 

592 
600 

0.021 
0.022 

0.1877 
0.2060 

183 

0.071 

0.072 

2.1451 
2  .  2059 

608 

617 

0.023 
0.024 
0.025 

0.2251 
0.2451 
0  .  2660 

200 
209 

0.073 
0.074 
0.075 

2.2676 
2  .  3302 
2.3936 

626 
634 

0,40 

0.026 
0.027 
0.028 

0.2877 
0.3102 
0.3336 

225 
234 

0.076 
0.077 
0.078 

2.4579 
2  .  5230 
2  .  5889 

o4o 
651 
659 

668 

0.029 
0.030 

0.3579 
0.3830 

251 
259 

0.079 
0.080 

2  .  6557 
2.7234 

677 
685 

0.031 
0.032 
0.033 
0.034 
0.035 
0.036 
0.037 
0.038 
0.039 

0.4089 
0.4357 
0.4634 
0.4919 
0.5213 
0.5515 
0  .  5826 
0.6145 
0.6472 

268 
277 
285 
294 
302 
311 
319 
327 
336 

0.081 
0.082 
0.083 
0.084 
0.085 
0.086 
0.087 
0.088 
0.089 

2.7919 
2.8613 
2.9315 
3  .  0026 
3.0745 
3.1472 
3  .  2208 
3.2953 
3.3706 

694 
702 
711 

719 
727 
736 
745 
753 
762 

0.040 

0.6808 

345 

0.090 

3.4468 

770 

0.041 

0.042 
0.043 
0.044 
0  .  045 
0.046 
0.047 
0.048 
0.049 
0.050 

0.7153 
0.7506 
0.7868 
0.8238 
0.8617 
0.9004 
0.9400 
0  .  9804 
1.0217 
1.0638 

353 
362 
370 
379 
387 
396 
404 
413 
421 
430 

0.091 
0.092 
0.093 
0.094 
0.095 
0.096 
0.097 
0.098 
0.099 
0.100 

3.5238 
3.6017 
3.6804 
3.7600 
3.8404 
3.9217 
4.0038 
4  .  0868 
4.1706 
4  .  2553 

779 
787 
796 
804 
813 
821 
830 
838 
847 
855 

316 


FLUIDITY  AND  PLASTICITY 


TABLE  X. — LENGTHS  OF  CAPILLARY  FOR  DIFFERENT  RADII  ASSUMING 
MAXIMUM  FLUIDITY  OF  500,   A   MINIMUM  PRESSURE  OF  50  G   PER 
SQUARE  CENTIMETER,  A  MINIMUM  TIME  OF  FLOW  OF  200  SEC.,  AND  A 

VOLUME    OF    FLOW    OF    4  ML.     I  =  -f*-%H« 


Radius  in 
centimeters 

Length    in 
centimeters 

Difference 

Radius   in 
centimeters 

Length  in 
centimeters 

Difference 

0.001 

0.00048 

0.051 

•3,254 

0.002 

0.00770 

0.052 

3,517 

263 
279 

0.003 

0  .  03896 

0.053 

3,796 

0.004 

0.12315 

0.054 

4,090 

294 

0.005 

0.3007 

0.055 

4,402 

•3OO 

0.006 

0.6234 

0.056 

4,731 

0 

0.007 
0.008 

1.155 
1.970 

"6!825' 
1    186 

0.057 
0.058 

5,080 
5,446 

o49 
366 

0.009 
0.010 

3.156 
4.811 

1  ;  655 
2.232 

0.059 
0.060 

5,830 
6,234 

384 
404 
429 

0.011 
0.012 

7.043 
9.978 

2.935 

3QA 

0.061 
0.062 

6,663 
7,110 

447 

0.013 
0.014 
0.015 
0.016 
0.017 
0.018 
0.019 
0.020 

13.74 
18.48 
24.35 
31.53 
40.18 
50.50 
62.68 
76.97 

.  yb 
4.74 
5.87 
7.18 
8.65 
10.32 
12.  18 
14.29 
16.60 

0.063 
0.064 
0.065 
0.066 
0.067 
0.068 
0.069 
0.070 

7,577 
8,072 
8,587 
9,126 
9,693 
10,285 
10,904 
1  1  ,  550 

467 
495 
515 
539 
567 
592 
619 
646 
674 

0.021 
0.022 
0.023 
0.024 
0.025 
0.026 

93.56 
112.7 
134.6 
159.6 
187.9 
219.9 

19.12 
21.94 
25.0 
28.3 
32.0 
35  6 

0.071 
0.072 
0.073 
0.074 
0.075 
0.076 

12,224 
12,926 
13,662 
14  ,  427 
15,221 
16,048 

702 
736 
765 
794 

827 

0.027 

255.7 

0.077 

16  ,  909 

861 

0.028 
0.029 

295.7 
340.3 

44ie 

48  4 

0.078 
0.079 

17,809 
18,737 

900 

928 

0.030 

389.7 

54^6 

0.080 

19,704 

1,004 

0.031 

444.3 

0.081 

20,708 

042 

0.032 

504.4 

60.  1 

0.082 

21,750 

'nan 

0.033 
0.034 
0.035 
0.036 
0.037 
0.038 
0.039 
0.040 

570.3 
643.0 
721.8 
808.0 
901.8 
,003.0 
,113.0 
,232.0 

65.  9 
72.7 
78.8 
86.2 
93.8 
101.2 
110.0 
119.0 
127.0 

0.083 
0.084 
0.085 
0.086 
0.087 
0.088 
0.089 
0.090 

22  ,  830 
23  ,  950 
25,111 
26,314 
27  ,  560 
28  ,  849 
30,182 
31,562 

,UoU 
,120 
,161 
,203 
,246 
,289 
,333 
,380 
,426 

0.041 
0.042 
0.043 
0.044 
0.045 
0.046 
0.047 
0.048 
0.049 
0.050 

,359.0 
,497.0 
,645.0 
,803.0 
,972.0 
2,154.0 
2,348.0 
2,554.0 
2,774.0 
3,007.0 

138.0 
148.0 
158.0 
169.0 
182.0 
194.0 
206.0 
220.0 
233.0 
247.0 

0.091 
0.092 
0.093 
0.094 
0.095 
0.096 
0.097 
0.098 
0.099 
0.100 

32,988 
34,462 
35  ,  985 
37  ,  559 
39,183 
40,859 
42,587 
44,371 
46,210 
48,106 

,474 
,523 
,574 
,624 
,676 
,728 
,784 
,839 
,896 

APPENDIX  A 


317 


and  XII,   without  any  preliminary  measurements.     Table  X 
will  then  be  used  as  already  indicated. 

TABLE  XI. — APPROXIMATE  FLUIDITIES  FOR  CONVENIENT  REFERENCE 


Substance 


Fluidity 


Castor  oil  at  20° 0.1 

Lard  oil  at  20° 1 

Sugar  solution  at  20°,  60  per  cent  by  weight 1 . 77 

Water  at  20°. 2 100.0 

Aliphatic  hydrocarbons  and  ethers  at  boiling  tem- 
perature   500. 

Carbon  dioxide  at  the  critical  state ...  5 , 000 . 


TABLE  XII. — RADII  LIMITS  FOR  DIFFERENT  FLUIDITY  MAXIMA  AND  l/r 
RATIOS 


Fluidity    maximum 

I 
r 

Radius  limits  in  centi- 
meters 

5,000.0 

600 

0.005  to  0.008 

500.0 

500 

0.010  to  0.015 

50.0 

400 

0.020  to  0.026 

5.0 

300 

0.040  to  0.045 

0.5 

200 

0.074  to  0.081 

Calibrating  an  instrument  whose  maximum  fluidity  is  below 
100  offers  difficulties  since  water  at  20°  is  excluded.  The  best 
suggestions  at  present  are:  a  40  per  cent  solution  by  weight  of 
ethyl  alcohol  in  water  at  0°,  fluidity,  14.0;  a  60  per  cent  sucrose 
solution  by  weight  at  20°,  fluidity  1.77;  freshly  distilled  aniline  at 
0.5°,  fluidity,  9.95. 

The  glass-blower  should  make  the  bulb  C  (and  K)  to  have  the 
shape  of  two  hollow  cones  placed  base  to  base,  in  order  to  secure 
good  drainage.  Bad  drainage  may  be  detected  by  inequality  in 
the  fluidity  as  determined  by  the  right  and  left  limbs  with  viscous 
liquids  at  the  higher  rates  of  flow.  For  very  viscous  liquids  the 
time  of  flow  must  be  increased  so  that  drainage  difficulties  may 
be  obviated.  Increasing  the  size  of  the  bulbs  is  no  advantage. 

The  bulbs  must  be  of  such  size  that  the  left  meniscus  will  not 


318  FLUIDITY  AND  PLASTICITY 

only  be  at  A  when  the  right  meniscus  is  at  (?,  but  it  should  be  at 
B  when  the  right  meniscus  is  at  J,  and  at  D  when  the  right  menis- 
cus is  at  L. 

The  ends  of  the  capillary  at  E  and  F  must  not  be  contracted 
in  under  any  circumstances,  and  as  far  as  practicable  the  ends 
should  not  be  expanded  into  a  trumpet  shaped  opening,  as  this 
may  seriously  affect  the  kinetic  energy  correction.  The  appro- 
priateness of  the  correction  already  given  may  be  tested  for  each 
instrument  by  using  a  given  liquid  at  a  variety  of  pressures. 

If  the  liquid  is  to  move  past  the  marks  B  and  D  with  a 
velocity  of  not  less  than  0.1  cm  per  second  when  the  time  of 
flow  is  200  seconds,  it  is  only  necessary  to  use  tubing  for 
that  part  of  the  instrument  whose  radius  is  not  more  than 
0.25  cm.  On  the  other  hand,  for  absolute  measurements, 
the  instrument  should  always  be  so  designed  that  the  resist- 
ance to  flow  outside  of  the  capillary  will  be  negligible. 
This  is  ascertained  as  follows:  Let  the  lengths  of  the  tubes 
B,  D,  L,  and  J,  Fig.  1,  be  in  all  L  and  their  radius  be  R. 
Then  R4/L  must  be  greater  than  1,000  r4/l  For  a  capillary 
whose  radius  is  0.012  cm,  R  must  be  at  least  0.07  cm  if  L  =  1. 
This  value  is  larger  than  is  commonly  supposed.  A  rule  which 
is  simple  but  will  cover  every  case  is  to  have  R  at  least  10  times 
the  radius  of  the  capillary. 


APPENDIX  B 
PRACTICAL  PLASTOMETRY 

The  measurement  of  the  flow  of  plastic  substances  resembles 
that  of  viscous  substances  in  most  respects,  but  since  plastic 
substances  do  not  drain  like  liquids,  it  is  convenient  to  measure 
the  volume  (or  weight)  of  substance  extruded.  For  this  purpose 
the  plastometer,  shown  in  Fig.  30,  p.  77,  has  been  designed  to 
replace  the  viscometer.  It  consists  of  a  top  A,  container  B,  and 
base  C  with  capillary  D  and  receiver  E.  The  top  consists  of  a 
square  plate  of  brass,  through  which  the  pressure  is  admitted  by 
means  of  a  copper  tube  F,  which  is  enlarged  at  the  end  to  make  it 
convenient  to  connect  with  the  pressure.  The  rubber  gaskets  H 
and  J  enable  one  to  make  the  apparatus  gastight,  when  the 
thumb-screws  are  screwed  down.  A  brass  pin  K,  brazed  into  the 
top,  passes  through  the  rubber  gasket  and  into  a  hole  in  the  body 
of  the  container.  On  the  opposite  side  of  the  container,  a  small 
copper  tube  passes  through  the  top,  through  the  rubber  gasket, 
and  into  a  hole  which  extends  all  of  the  length  of  the  container, 
and  into  the  lower  end  of  which  a  short  piece  of  hollow  copper 
tubing  is  affixed.  This  tube  in  turn  passes  through  the  second 
rubber  gasket  J  leading  into  the  base  C,  thus  affording  a  connec- 
tion between  the  atmosphere  and  the  receiver  while  the  plas- 
tometer is  immersed  in  the  bath. 

The  receiver  is  made  of  glass  and  with  a  flat  bottom  so  that  it 
will  sit  upright.  It  is  held  in  position  by  means  of  the  rubber 
collar  M.  The  rod  G  is  attached  to  the  container  in  order  that 
the  plastometer  may  be  supported  by  the  frame  shown  in  Fig.  89. 

Through  the  base,  there  extends  the  capillary  tube  D,  whose 
ends  have  been  ground  off  flat  and  whose  dimensions  are  known. 
To  cement  the  capillary  in  position  it  is  cleaned  carefully  with 
chromic  acid  mixture  and  dried  without  touching  the  part  to  be 
soldered.  It  is  "tinned  over"  in  the  usual  manner  and  soldered 
in  place,  the  space  N  being  filled  with  the  alloy.  Two  parts 
bismuth,  two  parts  lead  and  one  part  tin  has  been  found  satis- 

319 


320  FLUIDITY  AND  PLASTICITY 

factory.     A  more  certain  method  is  to  platinize  the  glass  and 
then  solder  in  position  with  pure  tin  as  solder. 

To  determine  the  effect  upon  the  flow  of  changing  the  length 
and  radius  of  the  capillary,  at  least  four  capillaries  may  be 
required,  hence  it  is  convenient  to  keep  all  of  them  mounted 
continuously  in  duplicate  bases. 

Since  the  mercury  thread  gives  only  the  radius  of  a  cylinder 
which  would  have  the  same  volume  as  the  capillary,  the  true  aver- 
age radius  for  flow  purposes  requires  more  elaborate  estimation  if 
absolute  measurements  of  flow  are  to  be  made  and  any  consider- 
able accuracy  is  desired.  In  fact,  since  the  flow  varies  as  the 
fourth  power  of  the  radius  and  it  is  measurable  to  the  desired 
accuracy  only  with  difficulty,  it  may  be  said  that  this  is  the  most 
difficult  part  of  absolute  measurement. 

Whereas  one  will  seek  to  obtain  a  capillary  tube  which  is  a 
true  cylinder,  it  will  usually  be  slightly  elliptical.     In  this  case 
the  ratio  of  the  major  to  the  minor  axes  may  be  obtained  by 
measurements  of  the  photomicrograph  of  the  ends,  although 
several  other  methods  may  be  used.     From  this  ratio  and  the 
average  radius  obtained  by  means  of  the  mercury  calibration, 
the  actual  values  of  the  major  and  minor  axes  2B  and  2C  may 
be  calculated.1     It  follows  then  that 
4         2B3CS 
B2  +  C2' 

If  the  capillary  is  a  frustrum  of  a  true  cone, 


where  R*  and  R*  are  the  radii  of  the  two  ends.     If  the  capillary 
is  not  only  conical  but  elliptical  at  the  same  time, 


D   _   r< 

where  R3  and  R*  are  the  average  radii  of  the  two  ends,  e  =  jr+~n' 
and  2B  and  2C  are  the  mean  major  and  minor  axes. 

THE  MEASUREMENT  OF  PLASTICITY 

Until  the  pressure  is  admitted  the  flow  by  seepage  will  ordi- 
1  Cf.  RUCKER,  Phil.  Trans.  186A,  438  (1894)  and  KNIBBS,  ./.  and  Proc. 
Roy.  Soc.  New  South  Wales  29,  77  (1895);  30,  186  (1896). 


APPENDIX  B  321 

narily  be  extremely  slow.  It  is  possible  therefore  to  wipe  off  the 
end  of  the  capillary,  put  the  weighed  container  in  place,  admit 
the  pressure  for  a  known  interval  of  time,  touch  off  into  the 
container  any  material  still  adhering  to  the  capillary  and  weigh. 
From  the  weight  of  material,  the  volume  of  flow  may  be  cal- 
culated from  the  density  when  desired. 

There  is  however,  another  convenient  method  which  can  be 
used  when  the  material  comes  from  the  capillary  in  drops. 
The  observer  turns  on  the  pressure  and  simply  takes  the  time  of 
formation  of  a  convenient  number  of  drops,  making  no  weighing 
at  all.  Other  measurements  are  made  at  the  same  or  other 
pressures.  Finally  without  cleaning  off  the  end  of  the  capillary 
a  certain  number  of  drops  are  counted  off  into  a  weighed  receiver 
at  the  minimum  pressure  used  and  also  at  the  maximum  pressure 
used.  From  the  weight  of  a  drop  at  these  two  pressures,  one  can 
calculate  the  weight  of  a  drop  at  any  intermediate  pressure  pro- 
vided the  weight  is  a  linear  function  of  the  pressure.  By  this 
method  a  large  number  of  measurements  on  a  given  material  can 
be  completed  in  a  single  day  with  an  accuracy  of  0.3  per  cent. 
According  to  measurements  by  H.  D.  Bruce  the  weight  of  the 
drop  is  not  always  uniform  at  a  given  pressure. 

The  pressure  pi  delivered  to  the  plastometer  is  calculated  in 
the  manner  already  described  (page  299  et  seq.),  correcting  for  the 
temperature  of  the  liquid  in  the  manometer.  The  plastic 
material  exerts  a  hydrostatic  head  which  must  be  corrected  for 
as  follows. 

The  initial  head  in  the  container,  h,  may  be  measured  by  the 
use  of  a  straight,  slender  wire.  To  this  is  added  the  length  of 
the  capillary,  I,  hence  the  pressure  (h  +  l}p  added  to  p\,  gives 
the  corrected  pressure  p  to  be  used  in  calculating  the  plasticity. 
The  change  of  hydrostatic  head  in  subsequent  determinations 
may  be  ascertained  by  noting  the  volume  of  plastic  material 
which  has  accumulated  in  the  graduated  receiver.  In  this  case 
it  is  also  necessary  to  know  how  much  the  level  of  the  material 
in  the  container  is  lowered  by  the  loss  of  1  ml.  A  much  better 
plan  is  to  have  a  graduated  glass  tube  of  just  the  size  to  fit  into 
the  container,  and  open  at  both  ends,  cemented  into  the  container. 
Having  cut  away  portions  of  the  metal  of  the  container,  the  level 
of  the  material  within  may  be  read  directly. 
21 


322  FLUIDITY  AND  PLASTICITY 

In  the  measurements  of  plasticity  it  has  been  found  that 
high  pressures  give  data  which  may  be  handled  more  simply 
than  the  data  at  low  pressures.  But  a  multiple-tube  stabilizer 
to  give  two  atmospheres  of  pressure  is  both  complicated  and 
expensive,  hence  a  mercury  stabilizer  seems  desirable.  However 
a  mercury  stabilizer  was  not  used  at  first  because  as  soon  as  the 
pressure  became  great  enough  to  bubble  through  the  mercury  at 
all,  a  large  amount  of  gas  suddenly  came  off  causing  a  violent 
fluctuation  in  the  pressure.  This  intermittent  flow  of  air  is 
partly  due  to  the  failure  of  the  mercury  to  wet  the  tube  allowing 
a  continuous  air  channel  to  be  formed  over  a  considerable 
distance  between  the  mercury  and  the  tube.  This  difficulty 
can  be  overcome  by  the  amalgamation  of  the  tube  by  means  of 
sodium  amalgam.  A  further  difficulty  arose  from  the  necessity 
of  keeping  the  volume  of  gas  bubbling  through  the  stabilizer  as 
small  as  possible  while  maintaining  the  flow  continuously.  This 
trouble  was  completely  overcome  by  placing  a  Davis-Bourneville 
reducing  valve  at  the  point  C  of  the  apparatus  shown  in  Fig.  92, 
a  flow  indicator  just  between  the  needle-valve  D  and  the  pressure- 
reservoir  F,  and  another  flow  indicator  between  the  valve  E 
and  the  mercury  stabilizer. 

The  flow  indicator  consists  of  two  similar  vials  connected  by 
an  inverted  U-tube  leading  to  the  bottom  of  both  vials  through 
two-hole  rubber  stoppers.  A  little  glycerol  is  added  to  one  of  the 
vials  at  the  start  and  the  rate  of  bubbling  of  the  gas  through  the 
liquid  serves  to  indicate  the  direction  of  movement  of  the  gas 
as  well  as  its  velocity. 

The  mercury  stabilizer  consists  of  an  single  iron  tube  of  some 
25  mm  internal  diameter  into  which  leads  the  inner  tube  having 
a  diameter  of  5  mm  just  as  in  the  water  stabilizer.  The  outer 
tube  is  closed  at  the  bottom  by  means  of  a  cap  but  near  the 
bottom  a  side  tube  leads  off  for  the  attachment  of  a  stout  rubber 
tube  which  is  connected  in  turn  with  a  glass  receiver  of  about 
2  liters  capacity.  This  receiver  can  be  raised  and  lowered  and 
hung  on  stout  hooks  provided  for  the  purpose  at  frequent  vertical 
intervals.  In  order  to  change  from  one  pressure  to  another,  it  is 
necessary  for  mercury  to  be  added  to  or  taken  from  the  stabilizer. 
This  is  very  easily  accomplished  by  simply  raising  or  lowering 
the  receiver.  For  a  pressure  of  two  atmospheres  not  over  10  kg 


APPENDIX  R  323 

of  mercury  are  required.  Were  a  smaller  tube  used  for  the  outer 
tube  of  the  stabilizer,  less  mercury  would  be  required  but  the 
manipulation  might  be  less  convenient.  A  photograph  of  the 
plastometer  occording  to  the  latest  design  used  by  Mr.  H.  D. 
Bruce  is  reproduced  in  the  frontispiece. 

TREATMENT  OF  PLASTICITY  DATA 

The  data  may  be  analyzed  either  algebraically  or  graphically. 
The  formula  for  plastic  flow  through  a  capillary  tube  is 

where  /*  is  the  mobility,  and  /  the  friction  or  yield  value.  The 
shearing  force,  F  =  ~,'  ,  is  expressed  in  dynes  per  square  centi- 

meter and  the  pressure  P  is  expressed  in  grams  per  square  centi- 
meter. Since  the  kinetic  energy  is  generally  negligible  this 
becomes 

M  -  F~  (13) 

where  v  is  the  volume  of  flow  per  second  and  K  is  a  constant 

1  273 
whose  value  is  ~W~'    If  we  substitute  in  Eq.  (13)  the  values 

Fi,  v\  and  F2,  v2  from  two  observations  of  the  flow,  we  find  that 
m  FlVz-F*Vl 

V-i   —  Vi 

so  that  both  /*  and  /  are  readily  determined.  Since  however  the 
weight  of  flow  w  =  vp,  a  more  convenient  expression  for  the 
friction  is 


m 

wz  —  w\ 

The  friction  must  have  a  positive  value  for  all  plastic  substances 
and  the  value  should  be  constant  for  a  given  capillary  so  long  as 
seepage,  slipping,  et  cet.,  do  not  intervene. 

In  the  early  stages  of  the  development  of  the  subject,  the 
graphical  method  of  treatment  is  desirable  from  many  points  of 
view.  Plotting  the  weight  of  flow  in  grams  per  second  as  ordi- 
nates  and  the  shear  in  dynes  per  square  centimeter  as  abscissas, 
the  value  of  the  intercept  of  the  extrapolated  curve  gives  the  value 
of  the  friction  and  the  slope  of  the  curve  determines  the  mobility. 
The  curvature  indicates  to  what  degree  seepage,  et  cet.,  enter  in. 


APPENDIX  C 
TECHNICAL  VISCOMETERS 

Instruments  very  different  from  those  employed  in  scientific 
work  are  much  in  vogue  both  in  this  country  and  abroad  for 
industrial  purposes,  particularly  in  the  oil  industry.  Thus  we 
have  the  Engler  Viskosimeter  in  Germany,  the  Redwood  Vis- 
cometer  in  Great  Britian,  the  Saybolt  Viscosimeter  in  the  United 
States,  the  Barbey  Ixometre  in  France  and  a  host  of  others.  Most 
of  them  seem  to  have  been  devised  with  the  idea  in  mind  that  the 
time  of  flow  of  a  given  quantity  of  various  liquids  through  an 
opening  is  approximately  proportional  to  the  viscosity,  without 
much  regard  to  the  character  of  the  opening.  There  is  usually 
a  container  which  is  filled  to  a  certain  level  and  a  short  efflux  tube 
opening  into  the  air.  The  number  of  seconds  required  for  a  given 
quantity  of  liquid  to  flow  out  under  gravity  is  taken  as  an  indica- 
tion of  the  viscosity. 

As  it  was  gradually  realized  that  these  times  of  flow  were  not 
even  proportional  to  the  true  viscosities,  efforts  have  not  been 
wanting  to  reduce  the  times  of  flow  to  true  viscosities.  Since 
the  pressure  is  due  to  an  average  head  of  liquid  h,  the  pressure  is 
hgp  and  the  viscosity  formula  1,  p.  295,  may  be  written 

5  -  At  -  1 

P  t 

Having  obtained  the  values  of  the  constants  A  and  B  by  cali- 
brating the  viscometer  with  liquids  of  known  viscosity  it  appears 
possible  to  calculate  the  kinematic  viscosity  ij/p;  but  if  absolute 
viscosities  are  desired  it  is  necessary  to  make  a  supplementary 
determination  of  the  density  p.  Thus  elaborate  tables  and 
charts  have  been  devised  for  converting  Engler  "Degrees" 
(cf.  Ubbelohde  (1907)),  and  Redwood  (c/.Higgins(  191 3),  Herschel 
(1918)  or  Saybolt  "Seconds")  into  true  viscosities. 

The  widespread  use  of  the  Saybolt  viscometer  in  this  country 
makes  desirable  the  inclusion  here  of  the  specifications  for  its  use 
adopted  by  the  American  Society  for  Testing  Materials. 

324 


APPENDIX  C 


325 


"1.  Viscosity. — Viscosity  shall  be  determined  by  means  of  the 
Saybolt  Standard  Universal  Viscosimeter. 

"2.  Apparatus. — (a)  The  Saybolt  Standard  Universal  Viscos- 


Sectional  View 
Standard  Oil  Tube. 


A  Oil  Tube  Thermometer. 

B  Bath  Thermometer. 

C  Electric  Heater. 

D  Turn-table  Cover. 

E  Overflow  Cup. 

F  Turntable  Handles. 

6  Steam  Inlet  or  Outlet. 

H  Steam  U-Tube. 

J  Standard  Oil  Tube. 


K   Stirring  Paddles. 

L    Bath  Vessel. 

M  Electric  Heater  Receptacle. 

N   Outlet  Cork  Stopper. 

P   Gas  Burner. 

Q   Strainer. 

R   Receiving  Flask. 

S    Base  Block. 

T   Tube  Cleaning  Plunger. 


FIG.  96. — The  Saybolt  Universal  Viscometer. 

imeter  (see  Fig.  96)  is  made  entirely  of  metal.  The  standard  oil 
tube  J  is  fitted  at  the  top  with  an  overflow  cup  E,  and  the  tube  is 
surrounded  by  a  bath  L.  At  the  bottom  of  the  standard  oil  tube 
is  a  small  outlet  tube  through  which  the  oil  to  be  tested  flows 
into  a  receiving  flask  R,  whose  capacity  to  a  mark  on  its  neck  is 


326 


FLUIDITY  AND  PLASTICITY 


60  (±0.15)  cc.  The  lower  end  of  the  outlet  tube  is  enclosed 
by  a  larger  tube,  which  when  stoppered  by  a  cork  N,  acts  as  a 
closed  air  chamber  and  prevents  the  flow  of  oil  through  the  outlet 
tube  until  the  cork  is  removed  and  the  test  started.  A  looped 
string  is  attached  to  the  lower  end  of  the  cork  as  an  aid  to  its 
rapid  removal.  The  bath  is  provided  with  two  stirring  paddles 
K  and  operated  by  two  turn-table  handles  F.  The  temperatures 
in  the  standard  oil  tube  and  in  the  bath  are  shown  by  ther- 
mometers, A  and  B.  The  bath  may  be  heated  by  a  gas  ring 
burner  P,  steam  U-tube  H,  or  electric  heater  C.  The  standard 
oil  tube  is  cleaned  by  means  of  a  tube  cleaning  plunger  T,  and 
all  oil  entering  the  standard  oil  tube  shall  be  strained  through  a 
30-mesh  brass  wire  strainer  Q.  A  stop  watch  is  used  for  taking 
the  time  of  flow  of  the  oil  and  a  pipette,  fitted  with  a  rubber 
suction  bulb,  is  used  for  draining  the  overflow  cup  of  the  stand- 
ard oil  tube. 

"(b)  The  standard  oil  tube  J  should  be  standardized  by  the 
U.  S.  Bureau  of  Standards,  Washington,  D.  C.,  and  shall  conform 
to  the  following  dimensions: 


Dimensions 

Minimum, 
centimeters 

Normal, 
centimeters 

Maximum, 
centimeters 

Inside  diameter  of  outlet  tube.  .  . 
Length  of  outlet  tube 

0.1750 
1  215 

0.1765 
1  225 

0.1780 
1  235 

Height   of  overflow  rim   above 
bottom  of  outlet  tube  

12.40 

12.50 

12.60 

Diameter  of  container  of  stand- 

ard oil  tube  

2.955 

2.975 

2.995 

Outer  diameter  of  outlet  tube  at 

lower  end  

0.2S 

0.30 

0.32 

"3.  Method.— Viscosity  shall  be  determined  at  100°F  (37.8°C), 
130°F  (54.4°C),  or  210°F  (98.9°C).  The  bath  shall  be  held 
constant  within  0.25°F  (0.14°C)  at  such  a  temperature  as  will 
maintain  the  desired  temperature  in  the  standard  oil  tube.  For 
viscosity  determinations  at  100  and  130°F,  oil  or  water  may  be 
used  as  the  bath  liquid.  For  viscosity  determinations  at  210°F, 
oil  shall  be  used  as  the  bath  liquid.  The  oil  for  the  bath  liquid 
should  be  a  pale  engine  oil  of  at  least  350°F  flash-point  (open 


APPENDIX  C  327 

cup).  Viscosity  determinations  shall  be  made  in  a  room  free 
from  draughts,  and  from  rapid  changes  in  temperature.  All  oil 
introduced  into  the  standard  oil  tube,  either  for  cleaning  or  for 
test,  shall  first  be  passed  through  the  strainer. 

"To  make  the  test,  heat  the  oil  to  the  necessary  temperature 
and  clean  out  the  standard  oil  tube  with  the  plunger,  using  some 
of  the  oil  to  be  tested.  Place  the  cork  stopper  into  the  lower 
end  of  the  air  chamber  at  the  bottom  of  the  standard  oil  tube. 
The  stopper  should  be  sufficiently  inserted  to  prevent  the  escape 
of  air,  but  should  not  touch  the  small  outlet  tube  of  the  standard 
oil  tube.  Heat  the  oil  to  be  tested,  outside  the  viscometer,  to 
slightly  below  the  temperature  at  which  the  viscosity  is  to  be 
determined  and  pour  it  into  the  standard  oil  tube  until  it  ceases 
to  overflow  into  the  overflow  cup.  By  means  of  the  oil  tube 
thermometer  keep  the  oil  in  the  standard  oil  tube  well  stirred  and 
also  stir  well  the  oil  in  the  bath.  It  is  extremely  important  that 
the  temperature  of  the  oil  in  the  oil  bath  be  maintained  constant 
during  the  entire  time  consumed  in  making  the  test.  When  the 
temperature  of  the  oil  in  the  bath  and  in  the  standard  oil  tube  are 
constant  and  the  oil  in  the  standard  tube  is  at  the  desired  tem- 
perature, withdraw  the  oil  tube  thermometer;  quickly  remove  the 
surplus  oil  from  the  overflow  cup  by  means  of  a  pipette  so  that 
the  level  of  the  oil  in  the  overflow  cup  is  below  the  level  of  the  oil 
in  the  tube  proper;  place  the  60-ml  flask  in  position  so  that  the 
oil  from  the  outlet  tube  will  flow  into  the  flask  without  making 
bubbles;  snap  the  cork  from  its  position,  and  at  the  same  instant 
start  the  stop  watch.  Stir  the  liquid  in  the  bath  during  the  run 
and  carefully  maintain  it  at  the  previously  determined  proper 
temperature.  Stop  the  watch  when  the  bottom  of  the  meniscus 
of  the  oil  reaches  the  mark  on  the  neck  of  the  receiving  flask. 

"The  time  in  seconds  for  the  delivery  of  60  ml  of  oil  is  the 
Saybolt  viscosity  of  the  oil  at  the  temperature  at  which  the  test 
was  made." 

There  is  little  to  recommend  any  one  of  these  instruments 
except  their  wide  use  in  their  respective  countries.  They  .are 
inaccurate  and  in  the  case  of  viscous  oils  time-consuming.  With 
volatile  solvents  they  cannot  be  used  at  all  due  to  evaporation. 
The  greatest  source  of  error  in  the  technical  instruments  is  due 
to  poor  temperature  control.  The  bath  around  the  container  is 


328  FLUIDITY  AND  PLASTICITY 

small,  the  stirring  ineffective  and  the  end  of  the  efflux  tube  is 
exposed  to  the  air  In  making  duplicate  determinations  the 
liquid  flows  out  into  the  air  and  generally  cools  off,  so  the  bath  is 
raised  to  somewhat  above  the  desired  temperature  in  order  to 
bring  the  temperature  back  again  to  the  large  mass  of  oil  in  the 
container.  If  the  run  is  started  when  the  temperature  comes  to 
the  proper  point,  it  is  almost  impossible  to  prevent  it  going  up 
during  the  run. 

Another  important  source  of  error  arises  from  the  very  extra- 
ordinary kinetic  energy  corrections  encountered.  The  Engler 
instrument,  for  example,  is  normally  calibrated  with  water  at 
20°C  and  the  kinetic  energy  correction  amounts  to  over  90  per 
cent  of  the  total  energy  expended.  The  viscosity  in  this  case  has 
but  little  part  in  determining  the  rate  of  flow,  and  we  have  already 
seen  that  the  coefficient  (ra)  of  the  kinetic  energy  correction  is 
subject  to  some  uncertainty. 

Closely  connected  with  the  kinetic  energy  correction,  are  the 
difficulties  due  to  end  effects  and  possible  turbulence  which  are 
aggravated  in  short,  wide  tubes. 

It  is  difficult  to  adequately  clean  this  type  of  instrument  or  to 
tell  when  it  has  been  properly  cleaned.  The  liquids  readily 
absorb  dust,  moisture  and  other  impurities  from  the  air  and  they 
may  thus  undergo  loss  or  chemical  change.  Meissner  (1910)  has 
made  a  study  of  these  sources  of  error.  Effects  of  surface  tension 
at  the  end  of  the  capillary,  of  the  changing  level  of  liquid  in 
the  container,  of  slow  drainage  of  oil  down  the  side  of  the  receiving 
flask  are  found  to  be  small  sources  of  error.  With  the  Saybolt 
instrument,  the  flow  is  started  by  pulling  out  a  stopper  from  the 
hollow  cylinder  below  the  efflux  tube.  One  must  see  that  no 
liquid  accumulates  in  the  air  space  above  the  stopper. 

Instruments  embodying  the  principles  worked  out  by  Coulomb 
and  Couette  have  been  devised  by  Doolittle,  Stormer,  and  Mac- 
Michael.  In  the  Stormer  instrument  a  cylinder  is  rotated  by  the 
force  arising  from  a  falling  weight,  suspended  by  a  cord  carried 
over  a  pulley.  The  speed  varies  with  the  viscosity  of  the  liquid 
and  the  revolutions  per  minute  are  counted.  A  better  plan  is  the 
one  adopted  by  MacMichael  of  using  a  constant  speed,  imparted 
to  an  outside  cup  and  measuring  the  angle  of  torque  produced  in 
a  disk  supported  in  the  liquid  by  means  of  a  steel  wire.  The 


APPENDIX  C  329 

instrument  has  considerable  range,  for  wires  of  differing  diameters 
can  be  used  for  widely  differing  viscosities.  The  readings  are 
instantaneous  and  the  instrument  is  compact  and  easily  manipu- 
lated. The  most  troublesome  feature  of  this  type  of  instrument 
is  the  lack  of  constancy  in  the  supporting  wire.  It  is  neces- 
sary to  use  these  wires  with  considerable  care  and  to  calibrate 
frequently.  Since  the  corrections  of  the  instrument  are  not 
fully  understood,  the  calibrating  fluid  should  have  nearly  the 
same  viscosity  as  the  viscosity  to  be  measured  (cf,  Herschel 
(1920)). 

For  liquids  of  high  viscosity,  the  falling  sphere  method  is  used 
industrially.  If  the  containing  vessel  does  not  have  a  diameter 
at  least  10  times  that  of  the  ball,  a  correction  must  be  applied 
Sheppard  (1917).  The  method  is  admirably  adapted  for  abso- 
lute measurements,  but  usually  workers  have  felt  dependent 
upon  calibrating  liquids,  and  since  there  is  a  dearth  of  calibrating 
fluids  of  high  viscosity  liquids  are  often  used  in  which  the  velocity 
of  fall  is  too  great  for  the  strict  application  of  Stokes'  law  and  a 
correction  has  to  be  made.  Reproducible  liquids  of  high  viscosity 
which  have  been  accurately  determined  should  be  available  for 
the  industrial  requirements. 


APPENDIX  D 

The  measurements  of  Poiseuille,  being  somewhat  inaccessible 
but  of  great  practical  as  well  as  historical  interest  are  given 
in  detail  in  the  following  tables.  Comparative  values  of  the 
viscosity  of  water  by  various  observers  with  all  of  the  known 
corrections  made  are  given  in  Table  II.  Since  specific  viscosities 
are  often  used,  relative  to  water  at  different  temperatures,  we  give 
the  viscosity  for  water  for  every  degree  from  0  to  100  in  Table 
III,  and  in  Tables  IV  and  V  we  give  the  fluidities  of  alcohol-water 
solutions  and  sucrose-water  solutions  as  possible  calibration  fluids 
where  water  would  be  too  fluid.  For  changing  viscosities  to 
fluidities  the  table  of  reciprocals  (Table  VI)  is  very  convenient. 
To  get' the  reciprocal  of  a  number  such  as  1.007,  the  first  part 
of  the  table  is  not  very  convenient  on  account  of  the  large 
differences  used  in  interpolation.  If  however  one  uses  instead 
10  X  0. 10070  in  the  latter  part  of  the  table,  fifth  column,  p.  343, 
the  number  9.93  X 10-1  is  found  as  the  reciprocal  without  enter- 
polation.  The  part  of  the  table  from  10.0  to  15.0  may  also  be 
used  for  this  same  purpose,  in  which  case  the  reciprocal  of  10.07 
is  found  in  the  ninth  column. 

A  table  of  four-place  logarithms  (Table  VII)  are  included, 
and  are  often  sufficiently  exact,  since  viscosities  are  generally  not 
more  accurate  than  one  part  in  1,000. 


330 


APPENDIX  D 


331 


TABLE  I. — MEASUREMENTS  OF  POISEUILLE 


1 

1 

Diameter  of  capillary  in 
centimeters 

^ 

a 

05 

il 

"o 

S 
a 

•8 
£ 

1 

^•j    >» 

"o  c 

ll 

Open  end 

Bulb  end 

O            i           w 

5  *> 

•H 

.S  g 

«  .0 

I  ! 

&    g 

|l 

ll 

1  -2 

Tempera 
perimen 

Volume  < 
at  10°C 

1  1  2 

•3*5 

111 

^°     « 

IQ 

I 

0 

o 

10 

385.870 

3,505.75 

i     ® 

jjj 

CO 

J5 

jjj 

o 

739.  114 

1,830.75 

0 

d 

d 

d 

d 

TO 

773.443 

1,750.00 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

0.6 

co 

774.291 

2,327.75 

5.0 

773.  400 

2,025.25 

10.0 

773.443 

1,750.00 

15.0 

773.  597 

1,528.00 

20.0 

775.  093 

1,344.50 

25.0 

774.  886 

1,195.00 

30.1 

775.  058 

1,067.50 

35.1 

774.451 

962.25 

40.1 

774.  354 

871.50 

45.0 

774.  827 

793.  25 

51.068 

20,085.0 

97.764 

10,361.0 

S 

8 

147.  834 

6,851.0 

Ai             w 

2 

Do. 

Do. 

0 

Do. 

193.  632 

5,233.0 

o 
d 

3 

d 

387.  675 

2,612.5 

738.  715 

1,372.5 

774.676 

1,308.0 

98.404 

6,921.0 

S 

0 

148.  320 

4,594.0 

A11            ~ 

o 

o           Do 

Do. 

Do. 

Do.            193.421 

3,515.0 

in 

d 

d 

387.  445 

1,757.0 

j 

774.810 

878.0 

0 

.0          i 

i  1   1 

§           Do. 

0 

Do. 

Do. 

Do. 

387.  520 
774.895 

880.0 

448.0 

(N         j          0 

d 

24.  661 

8,646.0 

49.  591 

4,355.0 

K5 

98.233 

2,194.0 

A™ 

Era 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

148.233 

1,455.0 

-< 

194.257 

1,116.0 

388.000 

571.0 

775.  160 

298.0 

23.638 

5,570.0 

49.  185 

2,699.0 

•0 

99.  221 

1,360.0 

Av 

S 

Do 

Do. 

Do. 

Do. 

Do. 

Do. 

148.  623 

918.  5 

d 

193.315 

718.0 

387.737 

381.0 

774.620 

207.0 

332 


FLUIDITY  AND  PLASTICITY 

TABLE  1.— (Continued) 


• 

g 

Diameter  of  capillary  in 

g 

ii 

-§ 

"S 

centimeters 

S 

c 

'S 

•M 

£ 

**- 

^ 

s  £ 

"3  J 

"8 

'•5 

o 

*TJ 

'a  o 

a 

1 

Open  end 

Bulb  end 

§ 

J= 

e  i 

3  £ 

5  *> 

•s 

i 

Length  it 

1* 

S  * 

li 

11 

ll 

H 

Volume  ( 
at  10°C 

Ih 

•5-0 

III 

24.  753 

3,828.75 

50.001 

1,923.75 

K5 

99.343 

994.00 

AVI 

Is 

Do. 

Do. 

Do. 

Do 

Do. 

Do. 

148.618 

682.00 

d 

193.010 

537.  75 

387.  887 

291.50 

773.  790 

165.75 

4.783 

3,926.75 

6.204 

3,072.00 

12.  129 

1  ,  685.  50 

24.003 

974.25 

•j 

1 

49.040           571.75 

Av" 

o 

Do. 

Do. 

Do 

Do. 

Do 

Do. 

98.  832           348.  75 

148.475 

267.00 

193.501 

224.00 

387.972           144.00 

773.  717 

95.00 

•: 

<«                    ifl 

388.256 

4,103.5 

B 

§22 

S 

3              o              S 

•4                  a         1       <3i 

739.333 

2,156.0 

o 

0                 0 

o 

0 

*             777.  863 

2,060.0 

2 

0 

0 

0 

6 

e 

55.  286 

21,430.0 

97.  922 

12,079.0 

S          g 

148.  275 

7,981.5 

Bi 

i 

o 

o 

Do. 

Do. 

Do. 

Do. 

193.947 

6,100.0 

^ 

e 

387.  695 

3,052.0 

739.467  I   1,600.0 

774.891   I   1,526.5 

99.163  '  7,804.0 

i(5                   ^ 

B 

149.679  '   5,165.0 

B" 

Jo          2 

O)                 O 

o 

Do. 

Do. 

Do. 

Do. 

193.441   I  3,997.0 

•*             b 

6 

387.130      1,995.0 

774.796  !       999.0 

49.091 

7,471.0 

CO 

„ 

98.315 

3,729.0 

B"i 

12          2 

2 

Do. 

Do. 

Do. 

Do. 

148.571 

2,473.0 

S 

0 

0 

193.  877 

1,892.0 

*i 

0 

0 

388.100 

946.0 

774.880 

473.0 

APPENDIX  D 
TABLE  I. — (Continued) 


333 


_§ 

| 

Diameter  of  capillary  in                 , 

S 

^ 

-3  i 

1 

centimeters 

£ 

IM 

_n 

.i    * 

"S  a 

"o 

1 

1 

Open  end 

Bulb  end 

o 

!s 

S 

•s 

!l 

l! 

I 

! 
3 

1   * 

J-a 

**     CS 

.1.2 

!'! 

aj    C 

1  i 

1] 

Volume  ( 
at  10°C 

iii 

ill 

24.756 

5,543.0 

49.  857 

2,762.0 

0 

3 

2 

99.  214 

1,400.0 

BIV           §             5 

O 

Do. 

Do. 

Do. 

Do. 

149.082 

935.0 

d             d             d 

193.  194 

728.9 

387.  024 

375.0 

774.  540 

199.0 

| 

24.  290 

2,386.0 

49.  578 

1,193.0 

5 

S 

99.  139 

621.0 

•Bv             n             3 

0 

Do. 

Do. 

Do. 

Do. 

149.098 

428.0 

6            d      ,       d 

193.  130 

34.00 

387.024 

189.0 

I 

773.  282 

110.0 

!                                 J 

0.5 

774.048  j  2,816.75 

5.0 

774.047 

2,422.75 

6.0 

773.  848 

2,350.50 

1 

10.0 

774.030 

2,093.50 

M 

Mean    diam. 

15.2 

1^ 

774.070 

1,826.00 

'  C            § 

at  1  0°C 

20.0 

2 

774.  110 

1,612.75 

d 

0.  0  085 

25. 

N 

774.841 

1,427.50 

30. 

774.  503 

1,280.50 

35. 

774.574 

1,149.50 

40. 

774.  676 

1,042.50 

45. 

774.  678 

949.00 

S 

i 

10 

0 

0 

t~ 

385.158      4,210.00 

C 

0 

g 

1             -              § 

738.969 

2,192.00 

2 

1 

d 

d 

d 

o     ;    -. 

774.030      2,093.50 

52.257 

23,135.00 

98.411 

12,280.00 

s 

149.241 

8,098.00 

Ci 

» 

8 

Do. 

Do. 

Do. 

Do. 

Do             193.314 

6,250.00 

i^ 

d 

387.  562 

3,118.00 

738.  767 

1,636.00 

774.757 

1,560.50 

99.868 

7,  997!  00 

1C 

rf 

149.034 

5,362.00 

c« 

o 

§ 

8 

Do. 

Do. 

Do.     i    Do. 

193.  867 

4,117.00 

d 

d 

386.  915 

2,065.00 

i 

774.  563 

1,029.00 

334 


FLUIDITY  AND  PLASTICITY 


TABLE  I.  —  (Continued) 


£ 

Diameter  of  capillary  iu 

g 

^ 

•3  i 

1 

I 

centimeters 

<*- 

J 

3  & 

•3  a 

•8 

B 

8 

Open  end 

Bulb  end 

o 

1 

•a 

'S  g 

.s  I 

a  | 

! 

Length  ii 

jfl 

1* 

|1 

O      QQ 

|  1 

Tempera 
perimen 

Volume  ( 
at  10°'C 

jii 

•So 

111 

49.  702 

7,765.00 

0 

.0 

98.921 

3,899.00 

Cm 

5 

3 

8 

Do. 

Do. 

Do. 

Do. 

148.  303 

2,598.50 

d 

d 

193.  544 

1,994.00 

387.  157 

995.00 

774.  677 

498.00 

24.  791 

6,186.75 

1 

49.931 

3,073.00 

!            10 

98.  322 

1,559.75 

Civ      i      S 

Do 

Do. 

Do. 

Do. 

Do. 

Do. 

148.  795 

1,029.50 

—' 

194.  102 

788.00 

387.  191 

399.00 

774.  607 

203.00 

24.  192 

3,587.00 

50.506 

1,768.00 

10 

99.  102 

904.00 

i 

149.  119 

606.50 

Cv 

d 

Do. 

Do. 

Do. 

Do. 

Do. 

Do 

194.217 

470.00 

387.  237 

245.00 

773.  327 

131.50 

§ 

o     ;     o 

386.  247 

9,708.00 

D 

1 

i 

1 

1 

o        !      o             738.137 
~       i      «      |       773.970 

5,080.00 
4,846.00 

2 

d 

d 

d 

d 

d 

54.785    35,460.00 

S 

S 

55.  796 

34,798.00 

>o 

»s 

co 

99.  508 

19,517.00 

Z)i 

1 

i 

I 

8 

1 

0 

Do. 

149.219 

13,021.00 

d 

d 

d 

d 

192.  907 

10,071.00 

386.  555 

5,025.00 

774.617 

2,506.00 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

5.00 

Do. 

774.887 

2,898.50 

10.00 

774.617 

2,506.00 

15.00 

773.271 

2,199.00 

Mean 

diam. 

20.00 

774.  119 

1,928.00 

0.004 

40406 

25.05 

775.045 

1,713.75 

30.07. 

774.356 

1,532.50 

35.00 

675.429 

1,375.50 

40.00 

774.475 

,246.75 

45.10 

774.077 

,138.00 

APPENDIX  D 

TABLE  I. — (Continued) 


335 


1 

i 

Diameter  of  capillary  in 

§ 

^ 

•s  i 

0) 

centimeters 

L 

.S 

tf 

M 

0 

S 

* 

.Q 

•A     >. 

0    c 

•3 
s 

+3 

d 

i 

Open  end 

Bulb  end 

"o 

£ 

3   *> 

1 

•s 

1    1 

S    a 

i! 

I 

1 

•I  -a 

Ji 

-|.2 

1  i 

ll 

H 

Volume  ( 
at  10°C 

III 

"o  "o 

|I1 

98.917    10,149.00 

10 

s 

§ 

147.857      6.789.00 

D11 

- 

s 

1 

Do. 

Do. 

2 

Do.            193.485      5,178.00 

'        <N 

d 

d 

386.847 

2,589.50 

773.  985 

1,293.00 

! 

50.374      7,978.00 

to 

§ 

97.  124      4,136.00 

7)iii 

S 

Do. 

Do.          Do. 

Do. 

148.248 

2,706.00 

1 

8 

8 

192.707 

2,084.75 

d 

d 

Q 

387.419       1,038.00 

775.866           519.00 

23.884      5,479.00 

50.276      2,611.75 

i 

S 

tsi 

97.440       1,373.00 

Z)iv            w 

8 

8 

Do. 

Do.          Do. 

Do. 

147.  889          897.  50 

d 

o' 

d 

193.459           697.00 

387.062           349.00 

772.  117           176.00 

i       0             „ 

58.211    26,625.00 

|                                        \       „              ..               386.218      4,020.00 

£ 

8         S 

<*               737.829      2,103.00 

M 

d 

d             d 

d 

o              774.017      2,006.00 

j     0.50 

773.808      2,705.00 

5.00 

774.757      2,318.50 

Mean  ;  diam. 

10.00 

774.017 

2,006.00 

0.002938 

15.00 

773.  709 

1,756.75 

20.00 

773.  475 

1,547.25 

£ 

Do. 

Do. 

Do. 

Do. 

Do. 

25.10 

Do. 

774.081 

1,372.25 

30.05 

775.271 

1,227.25 

35.07 

774.  563 

1,102.50 

40.10 

775.  329 

997.75 

45.00 

774.635 

908.  75 

:         0                 n 

96.  693 

5,903.50 

8             S 

147.  588 

3,868.00 

El 

»o 

8         8        DO. 

Do.           10° 

Do. 

193.  100 

2,955.00 

d 

d             d 

386.  787 

1,469.00 

773.  880 

736.  75 

24.  301 

5,651.00 

49.994 

2,751.00 

Ell 

0 

Do. 

Do 

Do 

Do 

Do. 

Do. 

96.  123 

1,426.00 

IN 

148.  307 

925.00 

d 

193.  357 

707.00 

386.  852 

354.00 

773.223 

178.00 

336 


FLUIDITY  AND  PLASTICITY 


TABLE  I. — (Continued) 


1 

•s 

2 
5 

1 

Diameter  of  capillary  in 
centimeters 

ire  of  ex-  1 

bulb  in  cc 

*i 

Hux  of  vol- 
ilb  in  sec- 

Open  end 

Bulb  end 

O 

° 

2  « 

"o 

.s  = 

w  £ 

Q 

Length  ir 

11 

|1 

|1 

|| 

Tempera 
peri  men 

°  y 

!« 

fig 

•0*0 

III 

j 

126.  92 

1,623.00 

303.08 

678.00 

<N 

Q 

o 

667.42 

306.00 

F 

eg 

S 

S 

3 

°0 

g 

1.352.10 

150.00 

05 

o 

i 

o 

<§ 

35 

1,981.50 

104.00 

c§ 

d 

d 

d 

d 

j^ 

2,620.34 

78.00 

5,198.60 

40.25 

i 

10,462.08 

20.00 

8336 

.  1,295.00 

127.60 

838.00 

163.71 

657.00 

o 

o 

(N 

g 

328.82 

326.00 

Fi 

O»                             ^H 

Q 

0 

Do. 

661.29 

162.00 

0 

5 

0 

i 

2 

1,321.87 

81.50 

S 

d 

d 

d 

d 

1,981.76 

56.00 

2,626.58 

42.00 

5,210.27 

22.50 

10,459.10 

12.50 

81.83 

666.00 

\ 

163.97 

328.00 

331.51 

165.00 

>o 

664.36 

84.50 

Fii 

» 

Do 

Do 

Do. 

Do. 

Do. 

Do. 

1,323.58 

45.00 

OJ 

1,984.95 

31.00 

2,585.33 

25.00 

5,207.73 

15.00 

10,454.54 

9.00 

82.08 

345.00 

163.  78 

175.00 

329.  77 

91.00 

2          S 

660.97 

50.00 

Fm 

1 

I       i 

Do. 

Do. 

Do. 

Do. 

1,323.56 

28.00 

d             d 

1,983.20 

21.75 

2,590.76 

17.50 

5,160.07 

11.00 

10,456.45 

7.00 

82.36 

191.00 

163.  39 

104.00 

328.04 

59.75 

S 

g 

661.59 

35.00 

ftr 

1 

1 

I 

Do. 

Do. 

Do. 

Do. 

1,321.19 

21.75 

.            ~              "•". 
n     I     e           o 

1,985.00 

16.  75 

2,614.60 

13.50 

5,205.08 

9.00 

10,458.02 

6.00 

APPENDIX  D 


TABLE  I. — (Continued) 


337 


3 

2 

1 

Diameter  of  capillary  in 
centimeters 

i 

g 

S    *-> 

"I   " 

"8  _g 

"o 

§ 

•3 

1 

Open  end 

Bulb  end 

-3   •£ 

1 

"8 

s  1 

3  ~ 

i 

9 

i 

4* 

§| 

is 

j. 

II 

fig 

•8-8 
111 

Q 

j  s  - 

S  S 

s  « 

§   S         H  ft 

>  * 

£  s  "• 

P    3    0 

74.29 

114.00 

83.89 

130.00 

162.89 

63.00 

329.  39 

39.00 

pv 

s 

Do. 

Do. 

Do 

Do. 

Do. 

Do. 

653.  49 

25.00 

^ 

1,306.69 

16.00 

1,985.29 

13.00 

2,606.37 

10.75 

5,146.62 

7.50 

10,456.65 

5.00 

0          j          0 

§ 

1.087.200 

407.00 

G 

i       ^ 

M 

8.60 

SO 

1,586.340 

281.00 

o       ;       o 

§ 

8.70 

S 

2,084.060 

213.00 

0 

666 

6 

8.80 

J5 

2,602.300 

170.00 

18.70 

145.300      2,290.00 

18.90 

269.220      1,232.00 

18.90 

520.  240 

634.00 

§*" 

18.90 

1,019.870 

323.00 

G1 

Do. 

Do. 

Do. 

Do. 

18.80 

Do. 

2,014.160 

162.00 

s 

18.70 

3,437.360 

97.00 

18.80 

6,841.370 

48.75 

, 

18.80 

10,191.540 

33.00 

ON           1           0 

g 

18.95 

145.  300 

1,115.00 

S          2 

S 

19.25 

269.220 

597.00 

_ 

0 

0 

0 

i 

19.30 

Do. 

518.  940 

305.50 

0 

6 

6 

6 

6 

19.50 

1,019.670 

155.00 

19.50 

2,014.360 

79.75 

g 

c? 

11.00 

2,316.870      9,048.00 

// 

i 

g 

g 

11.00 

0.5 

3,837.000      5,438.00 

SO 

6 

6 

11.10 

6,117.600      3,460.00 

10.80 

3,850.160 

388.00 

§t 
* 

0 

§ 

10.90 

4,610.230 

319.00 

/ 

0 

8 

11.00 

1.0 

5,370.130 

267.00 

N 

6 

6 

11.00 

6,127.360 

235.00 

7.50 

6,130.080 

261.00 

so 

o 

11.00 

54.987 

8,590.00 

K 

1 

0 

S 

0 

11.00 
11.00 

1.0 

210.129 
419.645 

2,250.00 
1,125.75 

«o 

6 

6 

11.00 

835.  565 

565.00 

12.00 

1,576.000 

286.00 

338 


FLUIDITY  AND  PLASTICITY 
TABLE  I. — (Continued) 


1 

1 

Diameter  of  capillary  in 
centimeters 

8 

c 

OS 

a  ^ 

It 

"3  a 

•3 

1 

Open  end 

Bulb  end 

•s 

1  « 

11 

! 

.5 
1 

If 

|i 

ii 

§  .2 

Tempera 
perimen 

Volume 
at  10°C 

j|l 

•3  o  ^ 

11.00 

1.0 

2,338.376 

197.50 

2 

0 

11.00 

3,095.540 

154.00 

K 

I 

0 

11.00 
11.00 

3,856.939 
4,616.534 

123.00 
106.25 

s 

d 

0 

11.00 

5,376.534 

88.25 

11.00 

6,136.534 

77.50 

7.00 

6,136.534 

86.75 

o 

g 

0 

0 

| 

M 

8 

8 

8 

8 

8 

10.00 

8 

775.000 

1,240.00 

* 

c 

o 

0 

0 

o 

Mi 

§ 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

775.000 

84.50 

6     1 

APPENDIX  D 


339 


TT 


.— i    O   *O   LC    »O   Tf    00 

ddddddddd 


_  O   t~*-    1C    ^O    i-H    O3    00 

OOOOb-CO»O»O>O-<*'*T<'<*<COCOCOCOO1<N 

dddddddddddddddd 


i-H    CO   00    <N 


dddddddd 


£§82 


OiCOS^-^CDCOOOlMt^-COOO-^ 

CT>c<iicco-^i^TtiTtioo-^eo'*t^'-H 

t^COiOiOiO-^Tti-^COCOCOCOINC^I 

d  d  d  d  d  d  d  d  d  d  d  d  d  d 


(NrH-^iOOlOtNiCOiCON-COO 


Tt*     IO     l^     ^H 

CO    rH     05    00 

CO    CO    (M    d 


»C   Tt<    00    t-    CO    CO    00 


CO   «C 

d  d 


OJi-iO^OOOfNiC 
l>-U3CO'-HOOOOOt^-CO 

r-I^H^IrH^-ldddd 


t^OO-^COTfCDTtlCO 
%£% 


CJ    O    ^H    1C 
OO   OO   I>   CO 

d  d  d  d 


340 


FLUIDITY  AND  PLASTICITY 


TABLE  III. — FLUIDITY  AND  VISCOSITY  OF  WATEH  CALCULATED  BY  FORMULA' 
FOR  EVERY  DEGREE  BETWEEN  0°  AND  100°C 


Tem- 
pera- 
ture, 
°C 

Flu- 
idity 

Vis- 
cosity 
in  cp 

Tem- 
pera- 
ture, 
°C 

Flu- 
idity 

Vis- 
cosity 
in  cp 

Tem- 
pera- 
ture, 
°C 

Flu- 
idity 

Vis- 
cosity 
in  cp 

0 

55.80 

.7921 

33 

132  .  93 

0.7523 

67 

236  .  25 

0.4233 

1 

57.76 

.7313 

34 

135.66 

0.7371 

68 

239.57 

0.4174 

2 

59.78 

.6728 

35 

138.40 

0.7225 

69 

242.91 

0.4117 

3 

61.76 

.6191 

36 

141.15 

0.7085 

70 

246.26 

0.4061 

4 

63.80 

.5674 

37 

143.95 

0.6947 

71 

249  .  63 

0.4006 

5 

65.84 

.5188 

38 

146.76 

0.6814 

72 

253  .  02 

0.3952 

6 

67.90 

.4728 

39 

149.60 

0.6685 

73 

256.42 

0.3900 

7 

70.01 

.4284 

40 

152.45 

0.6560 

74 

259.82 

0.3849 

8 

72.15 

.3860 

41 

155.30 

0.6439 

75 

263.25 

0.3799 

9 

74.28 

.3462 

42 

158.20 

0.6321 

76 

266.67 

0.3750 

10 

76.47 

.3077 

43 

161.11 

0.6207 

77 

270.12 

0.3702 

11 

78.66 

.2713; 

44 

164.02 

0.6097 

78 

273  .  57 

0.3655 

12 

80.89 

.  2363 

45 

167.00 

0.5988 

79 

277.04 

0.3610 

13 

83.14 

.  2028 

46 

169.97 

0.5883 

80 

280.53 

0.3565 

14 

85.40 

.  1709; 

47 

172.95 

0.5782 

81 

284.03 

0.3521 

15 

87.69 

.1404 

48 

175.95 

0.5683 

82 

287.53 

0.3478 

16 

90.00 

.1111 

49 

178.95 

0.5588 

83 

291.03 

0.3436 

17 

92.35 

.  0828 

50 

182.00 

0.5494 

84 

294.54 

0.3395 

18 

94.71 

.0559 

51 

185.05 

0.5404 

85 

298.06 

0.3355 

19 

97.10 

.0299 

52 

188.14 

0.5315 

86 

301  .  63 

0.3315 

20 

99.50 

.0050 

53 

191.23 

0.5229 

87 

305.21 

0.3276 

20.20 

100.00 

i.ooool 

54 

194.34 

0.5146 

88 

308.78 

0.3239 

21 

101.94 

0.98101 

55 

197.45 

0.5064 

89 

312.35 

0.3202 

22 

104.40 

0.9579 

56 

200.62 

0.4985 

90 

315.92 

0.3165 

23 

106.86 

0.9358 

57 

203.78 

0.4907 

91 

319.53 

0.3130 

24 

109.38 

0.9142 

58 

206.95 

0.4832 

92 

323.13 

0.3095 

25 

111.91 

0.8937 

59 

210.13 

0.4759 

93 

326  .  74 

0.3060 

26 

114.45 

0.8737 

60 

213.33 

0.4688 

94 

330.38 

0.3027 

27 

117.03 

0.8545 

61 

216.54 

0.4618 

95 

334.01 

0.2994 

28 

119.62 

0.8360 

62 

219.80 

0.4550 

96 

337.65 

0.2962 

29 

122.25 

0.8180 

63 

223.07 

0.4483 

97 

341  .  30 

0.2930 

30 

124.89 

0.8007 

64 

226.34 

0.4418 

98 

344.96 

0.2899 

31 

127.54 

0.7840 

65 

229  .  64 

0.4355 

99 

348.63 

0.2868 

32 

130.22 

0.7679 

66 

232.94 

0.4293 

100 

352.30 

0.2838 

4>  =  2. 1482  { (t  -  8.435)  +  V8078.4  +  (t  -  8.435) 2}  -  120.    Cf.  p.  137. 


APPENDIX  D  341 

TABLE  IV. — FLUIDITY  OF  ALCOHOL-WATER  MIXTURES1 


Weight  percentage  of  ethyl  alcohol 

Tem- 

0 

10 

20 

30 

39 

40 

45 

50 

60 

70 

80 

90 

100 

ture 

Volume  percentage  of  ethyl  alcohol  at  25°C 

1 

0 

12.36 

24.09 

35.23 

44.92 

45.83 

50.94 

55.93 

65.56 

74.80 

83.59 

92.01     100 

0 

55.8 

30.2 

18.8 

14.4 

13.8 

14.0 

14.4 

15.2 

17.4 

21.0 

27.1 

36.6 

56.4 

5 

65.8 

38.8 

24.6 

18.9 

17.8 

17.9 

18.2 

19.0 

21.6 

25.6 

32.0 

43.3 

61.6 

in 

76.5 

45.9 

31.6 

24.7 

22.8 

22.8 

23.0 

23.9 

26.5 

30.6 

36.9 

47.6 

68.2 

15 

87.7 

55.8 

38.2 

30.7 

28.4 

28.3 

28.5 

29.1 

31.8 

36.1 

43.3 

55.5 

75.1 

20 

99.5 

65.0 

45.8 

36.9 

34.7 

34.4 

34.7 

34.8 

37.4 

42.2 

49.8 

62.1 

83.3 

25 

111.9 

75.6 

55.1 

45.9 

42.5 

42.5 

41.9 

41.7 

44.6 

49.1 

57.2 

70.2 

91.2 

30 

124.9 

86.2 

64.4 

53.4 

50.0 

49.4 

49.5 

49.6 

51.9 

56.6 

65.3 

78.2 

99.7 

35 

138.4 

99.4 

75.1 

63.3 

58.6 

58.3 

57.7 

58.0 

60.1 

65.4 

73.8 

87.2 

109.4 

40 

152.4 

110.2 

86.2 

73.1 

67.9 

67.5 

66.9 

66.7 

69.1 

74.4 

83.1 

96.6 

119.9 

45 

167.0 

123.2 

98.5 

84.1 

77.9 

77.6 

76.5 

77.3 

78.7 

84.1 

92.5 

106.5 

130.8 

50 

182.0 

136.3  110.2 

95.2 

89.0 

88.3 

87.1 

86.6 

88.7 

94.2 

103.3 

117.9 

142  .  5 

55 

197.4 

150.9  122.9 

107.6 

100.7 

100.2 

98.4 

98.0 

100.3 

106.0 

115.3 

130.8 

155.2 

60 

213.3 

164.3  135.8 

119.9 

113.0 

112.0 

110.3 

109.5 

110.8 

116.8 

126.7 

142.1 

168.9 

65 

229.6 

180.5  150.  l!l33.0 

125.3 

124.7 

122.6 

122.3 

124.1 

130.6 

140.7 

156.0 

181.5 

70 

246.3 

194.5  164.5  146.4 

138.0 

137.5 

135.2 

135.1 

137.2 

143.9 

153.9 

169.9 

198.6 

75 

263.2 

210.21178.8  160.3  151.5 

150.8 

148.9 

148.7 

150.8 

157.1 

166.6 

183.0 

212.5 

80 

280.5 

232.7 

198.lll76.4 

167.1 

166.5 

164.1 

163.5 

165.7 

TABLE  V. — SUCROSE  SOLUTIONS,  BINGHAM  AND  JACKSON 


Tem- 

Percentage sucrose  by 
weight 

Tem- 

Percentage sucrose  by 
weight 

pera- 

pera- 

ture 

1 

ture 

0 

20 

40          60 

0 

20 

40 

60 

0 

55.91 

26.29 

6.77    0.42 

55 

197.16 

113.12 

45.06 

8.57 

5 

65.99 

31.71 

8.65    0.64 

60 

212.72 

123.79 

50.47 

10  17 

10 

76.56 

37.71 

10.21 

0.91 

65 

229.41 

134.81 

56.24 

11.99 

15 

87.67 

44.11 

13.39 

1.34 

70 

246.18 

145.97 

62.17 

13.98 

20 

99.54 

51.02 

16.13 

1.77 

75 

263.57 

157.56 

68.41 

16.12 

25 

111.84 

58.69 

19.28 

2.28 

80 

281.21 

169.53 

74.96 

18.51 

30 

124.70 

66.51 

22.82 

2.96 

85 

299.31 

181.80 

81.92 

21.14 

35 

138.79 

75.12 

26.58 

3.77 

90 

317.87  

89.06 

24.07 

40 

153.07 

83.82 

30.78 

4.70 

95 

335.46  

96.41 

26.85 

45 

167.84 

93.42 

35.13 

5.82 

100 

354.49  

104.11 

29.96 

50 

181.92 

103  .  07 

40.05 

7.14 

1  Values  given  are  the  weighted  average  of  those  of  Stephan  (1862),  Pagliani  and  Batelli 
(1885),  Traube  (1886),  Noack  (1886)  and  Bingham  and  Thomas  (1913). 


342 


FLUIDITY  AND  PLASTICITY 
TABLE  VI. — RECIPROCALS 


No.' 


i  Dif. 


1.0 

1.0000 

9901 

9804 

9709 

9615 

9524 

9434 

9346 

9259 

9174 

1.1 

0.9091 

9009 

8929 

8850 

8772 

8696 

8621 

8547 

8475 

8403 

1.2 

8333 

8264 

8197 

8130 

8065 

8000 

7937 

7874 

7813 

7752 

1.3 

7692 

7634 

7576 

7519 

7463 

7407 

7353 

7299 

7246 

7194 

1.4 

7143 

7092 

7042 

6993 

6944 

6897 

6849 

6803 

6757 

6711 

1.5 

0.  6667 

6623 

6579 

6536 

6494 

6452 

6410 

6369 

6329 

6289 

1.6 

6250 

6211 

6173 

6135 

6098 

6061 

6024 

5988 

5952 

5917 

1.7 

5882 

5848 

5814 

5780 

5747 

5714 

5682 

5650 

5618 

5587 

1.8 

5556 

5525 

5495 

5464 

5435 

5405 

5376 

5348 

5319 

5291 

1.9 

5263 

5236 

5208 

5181 

5155 

5128 

5102 

5076 

5051 

5025 

2.0 

0.5000 

4975 

4950 

4926 

4902 

4878 

4854 

4831 

4808 

4785 

2.1 

4762 

4739 

4717 

4695 

4673 

4651 

4630 

4608 

4587 

4566 

2.2 

4545 

4525 

4505 

4484 

4464 

4444 

4425 

4405 

4386 

4367 

2.3 

4348 

4329 

4310 

4292 

4274 

4255 

4237 

4219 

4202 

4184 

2.4 

4167 

4149 

4132 

4115 

4098 

4082 

4065 

4049 

4032 

4016 

2.5 

0.4000 

3984 

3968 

3953 

3937 

3922 

3906 

3891 

3876 

3861 

2.6 

3846 

3831 

3817 

3802 

3788 

3774 

3759 

3745 

3731 

3717 

2.7 

3704 

3690 

3676 

3663 

3650 

3636 

3623 

3610 

3597 

3584 

2.8 

3571 

3559 

3546 

3534 

3521 

3509 

3496 

3484 

3472 

3460 

2.9 

3448 

3436 

3425 

3413 

3401 

3390 

3378 

3367 

3356 

3344 

3.0 

0  3333 

3322 

3311 

3300 

3289 

3279 

3268 

3257 

3247 

3236 

3.1 

'3226 

3215 

3205 

3195 

3185 

3175 

3165 

3155 

3145 

3135 

3.2 

3125 

3115 

3106 

3096 

3086 

3077 

3067 

3058 

3049 

3040 

3.3 

3030 

3021 

3012 

3003 

2994 

2985 

2976 

2967 

2959 

2950 

3.4 

2941 

2933 

2924 

2915 

2907 

2899 

2890 

2882 

2874 

.  2865 

3.5 

0.  2857 

2849 

2841 

2833 

2825 

2817 

2809 

2801 

2793 

2786 

• 

3.6 

2778 

2770 

2762 

2755 

2747 

2740 

2732 

2725 

2717 

2710 

3.7 

2703 

2695 

2688 

2681 

2674 

2667 

2660 

2653 

2646 

2639 

7 

3.8 

2632 

2625 

2618 

2611 

2604 

2597 

2591 

2584 

2577 

2571 

3.9 

2564 

2558 

2551 

2545 

2538 

2532 

2525 

2519 

2513 

2506 

4.0 

0.2500 

2494 

2488 

2481 

2475 

2469 

2463 

2457 

2451 

2445 

• 

4.1 

2439 

2433 

2427 

2421 

2415 

2410 

2404 

2398 

2392 

2387 

4.2 

2381 

2375 

2370 

2364 

2358 

2353 

2347 

2342 

2336 

2331 

4.3 

2326 

2320 

2315 

2309 

2304 

2299 

2294 

2288 

2283 

2278 

4.4 

2273 

2268 

2262 

2257 

2252 

2247 

2242 

2237 

2232 

2227 

• 

4.5 

0.  2222 

2217 

2212 

2208 

2203 

2198 

2193 

2188 

2183 

2179 

4.6 

2174 

2169 

2165 

2160 

2155 

2151 

2146 

2141 

2137 

2132 

4.7 

2128 

2123 

2119 

2114 

2110 

2105 

2101 

2096 

2092 

2088 

4.8 

2083 

2079 

2075 

2070 

2066 

2062 

2058 

2053 

2049 

2045 

4.9 

2041 

2037 

2033 

2028 

2024 

2020 

2016 

2012 

2008 

2004 

5.0 

0.2000 

1996 

1992 

1988 

1984 

1980 

1976 

1972 

1969 

1965 

4 

5.1 

1961 

1957 

1953 

1949 

1946 

1942 

1938 

1934 

1931 

1927 

5.2 

1923 

1919 

1916 

1912 

1908 

1905 

1901 

1898 

1894 

1890 

5.3 

1887 

1883 

1880 

1876 

1873 

1869 

1866 

1862 

1859 

1855 

5.4 

1852 

1848 

1845 

1842 

1838 

1835 

1832 

1828 

1825 

1821 

5.5 

0.1818 

1815 

1812 

1808 

1805 

1802 

1799 

1795 

1792 

1789 

• 

5.6 

1786 

1783 

1779 

1776 

1773 

1770 

1767 

1764 

1761 

1757 

5.7 

1754 

1751 

1748 

1745 

1742 

1739 

1736 

1733 

1730 

1727 

5.8 

1724 

1721 

1718 

1715 

1712 

1709 

1706 

1704 

1701 

1698 

5.9 

1695 

1692 

1689 

1686 

1684 

1681 

1678 

1675 

1672 

1669 

APPENDIX  D 


343 


TABLE  VI. — RECIPROCALS  (Continued) 


No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Dif. 

6.0 

0.  16667 

16639 

16611 

16584 

16556 

16529 

16502 

16474 

16447 

16420 

27 

6.1 

16393 

16367 

16340 

16313 

16287 

16260 

16234 

16207 

16181 

16155 

28 

6.2 

16129 

16103 

16077 

16051 

16026   16000 

15974 

15949 

15924 

15898 

26 

6.3 

15873 

15848 

15823 

15798 

15773 

15748 

15723 

15699 

15674 

15649 

25 

6.4 

15625 

15601 

15576 

15552 

15528 

15504 

15480 

15456 

15432 

15408 

24 

6.5 

0.  15385 

15361 

15337 

15314 

15291 

15267 

15244 

15221 

15198 

15175 

23 

6.6 

15152 

15129 

15106 

15083 

15060 

15038 

15015 

14992 

14970 

14948 

23 

6.7 

14925 

14903 

14881 

14859 

14837 

14815 

14793 

14771 

14749 

14728 

22 

6.8:   14706 

14684 

14663 

14641 

14620 

14599 

14577 

14556 

14535 

14514 

21 

6.9 

14493 

14472 

14451 

14430 

14409 

14388 

14368 

14347 

14327 

14306 

21 

7.00.14286 

14265 

14245 

14225 

14205 

14184 

14164 

14144 

14124 

14104 

20 

7.1 

14085 

14065 

14045 

14025 

14006 

13986 

13966 

13947 

13928 

13908 

7.2 

13889 

13870 

13850 

13831 

13812 

13793 

13774 

13755 

13736 

13717 

19 

7.  3   13699 

13680 

13661 

13643 

13624 

13605 

13587 

13569 

13550 

13532 

7.4   13514 

13495 

13477 

13459 

13441 

13423 

13405 

13387 

13369 

13351 

18 

7.  5  0.  13333 

13316 

13298 

13280 

13263 

13245 

13228 

13210 

13193 

13175 

7.6   13158 

13141 

13123 

13106 

13089 

13072 

13055 

13038 

13021 

13004 

17 

7.7   12987 

12970 

12953 

12937 

12920 

12903 

12887 

12870 

12853 

12837 

7.81   12821 

12804 

12788 

12771 

12755 

12739 

12723 

12706 

12690 

12674 

18 

7.9 

12658 

12642 

12626 

12610 

12594 

12579 

12563 

12547 

12531 

12516 

8.0 

0.  12500 

12484 

12469 

12453 

12438 

12422 

12407 

12392 

12376 

12361 

8.  1 

12346 

12330 

12315 

12300 

12285 

12270 

12255 

12240 

12225 

12210 

15 

8.2 

12195 

12180 

12165 

12151 

12136 

12121 

12107 

12092 

12077 

12063 

8.3 

12048 

12034 

12019 

12005 

11990 

11976 

11962 

11947 

11933 

11919 

8.4 

11905 

11891 

11876 

11862 

11848 

11834 

11820 

11806 

11792 

11779 

14 

8  50.  11765 

11751 

11737 

11723 

11710 

11696 

11682 

11669 

11655 

11641 

8.6   11628 

11614 

11601 

11587 

11574 

11561 

11547 

11534 

11521 

11507 

8.7!   11494 

11481 

11468 

11455 

11442 

11429 

11416 

11403 

11390 

11377 

18 

8  8   11364 

11351 

11338 

11325 

11312 

11299 

11287 

11274 

11261 

11249 

8.9   11236 

11223 

11211 

11198 

11186 

11173 

11161 

11148 

11136 

11123 

9.00.  11111 

11099 

11086 

11074 

11062 

11050 

11038 

11025 

11013 

11001 

9  1   10989 

10977 

10965 

10953 

10941 

10929 

10917 

10905 

10893 

10881 

12 

9.  2   10870 

10858 

10846 

10834 

10823 

10811 

10799 

10787 

10776 

10764 

9.  3   10753 

10741 

10730 

10718 

10707 

10695 

10684 

10672 

10661 

10650 

9.4j  10638 

10627 

10616 

10604 

10593 

10582 

10571 

10560 

10549 

10537 

9.  5  0.  10526 

10515 

10504 

10493 

10482 

10471 

10460 

10449 

10438 

10428 

11 

9.6'   10417 

10406 

10395 

10384 

10373 

10363 

10352 

10341 

10331 

10320 

9.  71  10309 

10299 

10288 

10277 

10267 

10256 

10246 

10235 

10225 

10215 

9.8   10204 

10194 

10183 

10173 

10163 

10152 

10142 

10132 

10121 

10111 

9.9   10101 

10091 

10081 

10070 

10060 

10050 

10040 

10030 

10020 

10010 

10.00.10000 

9990 

9980 

9970 

9960 

9950 

9940 

9930 

9921 

9911 

10 

10.  11  09901 

9891 

9881 

9872 

9862 

9852 

9843 

9833 

9823 

9814 

10.  2   9804 

9794 

9785 

9775 

9766 

9756 

9747 

9737 

9728 

9718 

10.  3   9709 

9699 

9690 

9681 

9671 

9662 

9653 

9643 

9634 

9625 

10.4   9615 

9606 

9597 

9588 

9579 

9569 

9560 

9551 

9542 

9533 

10.  5  0.  09524 

9515 

9506 

9497 

9488 

9479 

9470 

9461 

9452 

9443 

9 

10.6 

9434 

9425 

9416 

9407 

9398 

9390 

9381 

9372 

9363 

9355 

10.7 

9346 

9337 

9328 

9320 

9311 

9302 

9294 

9285 

9276 

9268 

10.8 

9259 

9251 

9242 

9234 

9225 

9217 

9208 

9200 

9191 

9183 

10.9 

9174 

9166 

9158 

9149 

9141 

9132 

9124 

9116 

9107 

9099 

11.00.09091 

9083 

9074 

9066 

9058 

9050 

9042 

9033 

9025 

9017 

11.1    9009 

9001 

8993 

8985 

8977 

8969 

8961 

8953 

8944 

8937 

11.2   8929 

8921 

8913 

8905 

8897 

8889 

8881 

8873 

8865 

8857 

11.3   8850 

8842 

8834 

8826 

8818 

8811 

8803 

8795 

8787 

8780 

11.4    8772 

8764 

8757 

8749 

8741 

8734 

8726 

8718 

8711 

8703 

11.50.08696 

8689 

8681 

8673 

8666 

8658 

8650 

8643 

8636 

8628 

11.6   8621 

8613 

8606 

8598 

8591 

8584 

8576 

8569 

8562 

8554 

11.7   8547 

8540 

8532 

8525 

8518 

8511 

8503 

8496 

8489 

8482 

11.8   8475 

84  eV 

8460 

8453 

8446 

8439 

8432 

8425 

8418 

8410 

11.9   8403 

8396 

8389 

8382 

8375 

8368 

8361 

8354 

8347 

8340 

344 


FLUIDITY  AND  PLASTICITY 
TABLE  VI. — RECIPROCALS  (Continued) 


No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Dif. 

12.00.08333 

8326 

8320 

8312 

8306 

8299 

8292 

8285 

8288 

8271 

12.1    8264 

8258 

8251 

8244 

8237 

8230 

8224 

8217 

8210 

8203 

12.21   8197 

8190 

8183 

8177 

8170 

8163 

8157 

8150 

8143 

8137 

7 

12.3!   8130 

8124 

8117 

8110 

8104 

8097 

8091 

8084 

8078 

8071 

12.4 

8064 

8058 

8052 

8045 

8039 

8032 

8026 

8019 

8013 

8006 

12.50.08000 
12.  6    7936 

7994 
7930 

7987 
7924 

7981 
7918 

7974 
7912 

7968 
7905 

7962 

7899 

7955 
7893 

7949 

7886 

7942 

7880 

12.  7    7874 

7868 

7862 

7856 

7849 

7843 

7837 

7831 

7825 

7819 

12.8   7812 

7806 

7800 

7794 

7788 

7782 

7776 

7770 

7764 

7758 

12.9   7752 

7746 

7740 

7734 

7728 

7722 

7716 

7710 

7704 

7698 

13.00.07692 

7686 

7681 

7675 

7669 

7663 

7657 

7651 

7646 

7640 

• 

13.  1    7634 

7628 

7622 

7616 

7610 

7605 

7599 

7593 

7587 

7581 

13.2 

7576 

7570 

7564 

7559 

7553 

7547 

7542 

7536 

7530 

7524 

13.3 

7519 

7513 

7508 

7502 

7496 

7491 

7485 

7480 

7474 

7468 

13.4 

7463 

7457 

7452 

7446 

7441 

7435 

7430 

7424 

7418 

7413 

13.5 

0.07407 

7402 

7396 

7391 

7386 

7380 

7375 

7369 

7364 

7358 

13.6 

7353 

7348 

7342 

7337 

7332 

7326 

7321 

7315 

7310 

7305 

6 

13.7 

7299 

7294 

7289 

7283 

7278 

7273 

7268 

7262 

7257 

7252 

13.8 

7246 

7241 

7236 

7231 

7226 

7220 

7215 

7210 

7205 

7200 

13.9 

7194 

7189 

7184 

7179 

7174 

7169 

7164 

7158 

7153 

7148 

14.0 

0.07143 

7138 

7133 

7128 

7123 

7118 

7113 

7108 

7102 

7097 

14.1 

7092 

7087 

7082 

7077 

7072 

7067 

7062 

7057 

7052 

7047  | 

14.2 

7042 

7037 

7032 

7027 

7022 

7018 

7013 

7008 

7003 

0998  i 

14.3 

6993 

6988 

6983 

6978 

6974 

6969 

6964 

6959 

6954 

6949 

14.4 

6944 

6940 

6935 

6930 

6925 

6920 

6916 

6911 

6906 

6901 

14.5 

0.  06897 

6892 

6887 

6882 

687& 

6873 

6868 

6863 

6859 

6854 

14.6 

6849 

6845 

6840 

6835 

6931 

6826 

6821 

6817 

6812 

6807 

S 

14.7 

6803 

6798 

6793 

6789 

6784    6780 

6775 

6770 

6766 

6761 

14.8 

6757 

6752 

6748 

6743 

6739  i   6734 

6729 

6725 

6720 

6716 

14.9 

6711 

6707 

6702 

6698 

6693 

6689 

6684 

6680 

6676 

6671 

15.0 

0.06667 

6662 

6658 

6653 

6649 

6645 

6640 

6636 

6631 

6627 

15.1 

6623 

6618 

6614 

6609 

6605 

6601 

6596 

6592 

6588 

6583 

15.2 

6579 

6575 

6570 

6566 

6562 

6557 

6553 

6549 

6545 

6540 

15.3 

6536 

6532 

6527 

6523 

6519 

6515 

6510 

6506 

6502 

6498 

15.4 

6494 

6489 

6485 

6481 

6477 

6472 

6468 

6464 

6460 

6456 

15.5 

0.06452 

6447 

6443 

6439 

6435 

6431 

6427 

6423 

6419 

6414 

15.6 

6410 

6406 

6402 

6398 

6394 

6390 

6386 

6382 

6378 

6373 

15.7 

6369 

6365 

6361 

6357 

6353 

6349 

6345 

6341 

6337 

6333 

15.8 

6329 

6325 

6321 

6317 

6313 

6309 

6305 

6301 

6297 

6293 

4 

15.9 

6289 

6285 

6281 

6277 

6274 

6270 

6266 

6262 

6258 

6254 

16.0 

0.06250 

6246 

6242 

6238 

6234 

6231 

6227 

6223 

6219 

6215 

16.7 

6211 

6207 

6203 

6200 

6196 

6192 

6188 

6184 

6180 

6177 

16.2 

6173 

6169 

6165 

6161 

6158 

6154 

6150 

6146 

6143 

6139 

16.3 

6135 

6131 

6127 

6124 

6120 

6116 

6112 

6109 

6105 

6101 

16.4 

6097 

6094 

6090 

6086 

6083 

6079 

6075 

6072 

6068 

6064 

16.5 

0.06061 

6057 

6053 

6050 

6046 

6042 

6038 

6035 

6031 

6028 

16.6 

6024 

6020 

6017 

6013 

6010 

6006 

6002 

5999 

5995 

5992  ! 

18.7 

5988 

5984 

5981 

5977 

5973 

5970 

5966 

5963 

5959 

5956  i 

16.8 

5952 

5949 

5945 

5942 

5938    5935 

5931 

5928 

5924 

5921 

16.9 

5917 

5914 

5910 

5907 

5903  I   5900 

5896 

5893 

5889 

5886 

17.0 

0.05882 

5879 

5875 

5872 

5868 

5865 

5851 

5858 

5854 

5851 

17.1 

5847 

5844 

5841 

5838 

5834 

5861 

5828 

5824 

5821 

5817 

17.2 

5814 

5811 

5807 

5804 

5800 

5797 

5794 

5790 

5787 

5784 

a 

17.3 

5780 

5777 

5774 

5770 

5767 

5764 

5760 

5757 

5754 

5750 

17.4 

5747 

5744 

5741 

5737 

5734 

5731 

5727 

5724 

5721 

5718  | 

17.50.05714 

5711 

5708 

5704 

5701 

5698 

5695 

5692 

5688 

5685  ! 

17.  6|   5682 

5679 

5675 

5672 

5669 

5666 

5663 

5659 

5656 

5653  i 

17.7|   5650 

5647 

5613 

5640 

5637 

5634 

5631 

5627 

5624 

5621 

17.81   5618 

5615 

5612 

5609 

5605 

5602 

5599 

5596 

5593 

5590 

17.9   5587 

5583 

5580 

5577 

5574 

5571 

5568 

5565 

5562 

5559  i 

APPENDIX  D 
TABLE  VII. — LOGARITHMS 


345 


No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Dif. 

10    0000 

0043 

0086 

0128 

0170 

0212 

0253 

0294 

0334 

0374 

42 

11    0414 

0453 

0492 

0531 

0569 

0607 

0645 

0682 

0719 

0755 

as 

12    0792 

0828 

0864 

0899 

0934 

0969 

1004 

1038 

1072 

1106 

35 

13 

1139 

1173 

1206 

1239 

1271 

1303 

1335 

1367 

1399 

1430 

32 

14    1461 

1492 

1523 

1553 

1584 

1614 

1644 

1673 

1703 

1732 

ao 

15    17(11 

1790 

1818 

1847 

1875 

1903 

1931 

1959 

1987 

2014 

28 

16 

2041 

2068 

2095 

2122 

2148 

2175 

2201 

2227 

2253 

2279 

2ft 

17 

2304 

2330 

2355 

2380 

2405 

2430 

2455 

2480 

2504 

2529 

25 

18    2553 

2577 

2601 

2625 

2648 

2672 

2695 

2718 

2742 

2765 

28 

19  !  2788 

2810 

2833 

2856 

2878 

2900 

2923 

2945 

2967 

2989 

22 

20    3010 

3032 

3054 

3075 

3096 

3118 

3139 

3160 

3181 

3201 

21 

21 

3222 

3243 

3263 

3284 

3304 

3324 

3345 

3365 

3385 

3404 

20 

22    3424 

3444 

3464 

3483 

3502 

3522 

3541 

3560 

3579 

3598 

1» 

23  '  3617 

3636 

3655 

3674 

3692 

3711 

3729 

3747 

3766 

3784 

18 

24    3802 

3820 

3838 

3856 

3874 

3892 

3909 

3927 

3945 

3962 

18 

25    3979 

3997 

4014 

4031 

4048 

4065 

4082 

4099 

4116 

4133 

17 

26    4150 

4166 

4183 

4200 

4216 

4232 

4249 

4265 

4281 

4298 

1ft 

27    4314 

4330 

4346 

4362 

4378 

4393 

4409 

4425 

4440 

4456 

16 

28    4472 

4487 

4502 

4518 

4533 

4548 

4564 

4579 

4594 

4609 

IS 

29  '  4624 

4639 

4654 

4669 

4683 

4698 

4713 

4728 

4742 

4757 

16 

30  |  4771 

4786 

4800 

4814 

4829 

4843 

4857 

4871 

4886 

4900 

14 

31  i  4914 

4928 

4942 

4955 

4969 

4983 

4997 

5011 

5024 

5038 

32  1  5051 

5065 

5079 

5092 

5105 

5119 

5132 

5145 

5159 

5172 

33    5185 

5198 

5211 

5224 

5237 

5250 

5263 

5276 

5289 

5302 

IS 

34  i  5315 

5328 

5340 

5353 

5366 

5378 

5391 

5403 

5416 

5428 

35    5441 

5453 

5465 

5478 

5490 

5502 

5514 

5527 

5539 

5551 

36  :  5563 

5575 

5587 

5599 

5611 

5623 

5635 

5647 

5658 

5670 

12 

37   5682 

5694 

5705 

5717 

5729 

5740 

5752 

5763 

5775 

5786 

38  !  5798 

5809 

5821 

5832 

5843 

5855 

5866 

5877 

5888 

5899  i 

39   5911 

5922 

5933 

5944 

5955 

5966 

5977 

5988 

5999 

6010   11 

40 

6021 

6031 

6042 

6053 

6064 

6075 

6085 

6096 

6107 

6117 

41 

6128 

6138 

6149 

6160 

6170 

6180 

6191 

6201 

6212 

6222 

42    6232 

6243 

6253 

6263 

6274 

6284 

6294 

6304 

6314 

6325 

43 

6335 

6345 

6355 

6365 

6375 

6385 

6395 

6405 

6415 

6425 

10 

44 

6435 

6444 

6454 

6464 

6474 

6484 

6493 

6503 

6513 

6522 

48 

6532 

6542 

6551 

6561 

6571 

6580 

6590 

6599 

6609 

6618 

46 

6628 

6637 

6646 

6656 

6665 

6675 

6684 

6693 

6702 

6712 

47 

6721 

6730 

6739 

6749 

6758 

6767 

6776 

6785 

6794 

6803 

48 

6812 

6821 

6830 

6839 

6848 

6857 

6866 

6875 

6884 

6893 

» 

49 

6902 

6911 

6920 

6928 

6937 

6946 

6955 

6964 

6972 

6981 

r,o 

6990 

6998 

7007 

7016 

7024 

7033 

7042 

7050 

7059 

7067 

51 

7076 

7084 

7093 

7101 

7110 

7118 

7126 

7135 

7143 

7152 

52 

7160 

7168 

7177 

7185" 

7193 

7202 

7210 

7218 

7226 

7235 

53 

7243 

7251 

7259 

7267 

7275 

7284 

7292 

7300 

7308 

7316 

54    7324 

7332 

7340 

7348 

7356  '  7364 

7372 

7380 

7388 

7396    8 

346 


FLUIDITY  AND  PLASTICITY 
TABLE  VII. — LOGARITHMS  (Contimied) 


No. 

0 

1 

2 

3 

4      5 

6 

7 

8 

9 

Dif. 

55 

7404 

7412 

7419 

7427 

7435 

7443 

7451 

7459 

7466 

7474 

06 

7482 

7490 

7497 

7505 

7513 

7520 

7528 

7536 

7543 

7551 

57 

7559 

7566 

7574 

7582 

7589  |  7597 

7604 

7612 

7619 

7627 

68 

7634 

7642 

7649 

7657 

7664 

7672 

7679 

7686 

7694 

7701 

50 

7709 

7716 

7723 

7731 

7738 

7745 

7752 

7760 

7767 

7774 

M 

7782 

7789 

7796 

7803 

7810   7818 

7825 

7832 

7839 

7846 

61 

7853 

7860 

7868 

7875 

7882   7889 

7896 

7903 

7910 

7917 

82 

7924 

7931 

7938 

7945 

7952  !  7959 

7966 

7973 

7980 

7987 

7 

68 

7993 

8000 

8007 

8014 

8021 

8028 

8035 

8041 

8048 

8055 

64 

8062 

8069 

8075 

8082 

8089    8096 

8102 

8109 

8116 

8122 

65 

8129 

8136 

8142 

8149 

8156    8162 

8169 

8176 

8182 

8189 

66 

8195 

8202 

8209 

8215 

8222 

8228 

8235 

8241 

8248 

8254 

67 

8261 

8267 

8274 

8280 

8287 

8293 

8299 

8306 

8312 

8319 

68 

8325 

8331 

8338 

8344 

8351 

8357 

8363 

8370 

8376 

8382 

60 

8388 

8395 

8401 

8407 

8414 

8420 

8426 

8432 

8439 

8445 

70 

8451 

8457 

8463 

8470 

8476 

8482 

8488 

8494 

8500 

8506  ' 

71 

8513 

8519 

8525 

8531 

8537 

8543 

8549 

8555 

8561 

8567 

72 

8573 

8579 

8585 

8591 

8597 

8603 

8609 

8615 

8621 

8627   « 

73 

8633 

8639 

8645 

8651 

8657, 

8663 

8669 

8675 

8681 

8686 

74 

8692 

8698 

8704 

8710 

8716 

8722 

8727 

8733 

8739 

8745 

75 

8751 

8756 

8762 

8768 

8774 

8779 

8785 

8791 

8797 

8802 

71 

8808 

8814 

8820 

8825 

8831 

8837 

8842 

8848 

8854 

8859 

77 

8865 

8871 

8876 

8882 

8887 

8893 

8899 

8904 

8910 

8915 

78 

8921 

8927 

8932 

8938 

8943 

8949 

8954 

8960 

8965 

8971 

70 

8976 

8982 

8987 

8993 

8998  i  9004 

9009 

9015 

9020 

9025 

so 

9031 

9036 

9042 

9047 

9053  :  9058 

9063 

9069 

9074 

9079 

81 

9085 

9090 

9096 

9101 

9106  !  9112 

9117 

9122 

9128 

9133 

82 

9138 

9143 

9149 

9154 

9159  !  9165 

9170 

9175 

9180 

9186 

83 

9191 

9196 

9201 

9206 

9212   9217 

9222 

9227 

9232 

9238  | 

84 

9243 

9248 

9253 

9258 

9263   9269 

9274 

9279 

9284 

9289  i 

u 

9294 

9299 

9304 

9309 

9315   9320 

9325 

9330 

9335 

9340  ! 

86 

9345 

9350 

9355 

9360 

9365   9370 

9375 

9380 

9385 

9390  \ 

87 

9395 

9400 

9405 

9410 

9415  j  9420 

9425 

9430 

9435 

9440   8 

88 

9445 

9450 

9455 

9460 

9465  i  9469 

9474 

9479 

9484 

9489  1 

80 

9494 

9499 

9504 

9509 

9513 

9518 

9523 

9528 

9533 

9538 

00 

9542 

9547 

9552 

9557 

9562 

9566 

9571 

9576 

9581 

9586  : 

91 

9590 

9595 

9600 

9605 

9609 

9614 

9619 

9624 

9628 

9633 

02 

9638 

9643 

9647 

9652 

9657 

9661 

9666 

9671 

9675 

9680 

93 

9685 

9689 

9694 

9699 

9703 

9708 

9713 

9717 

9722 

9727  I 

04 

9731 

9736 

9741 

9745 

9750 

9754 

9759 

9763 

9768 

9773  ! 

06 

9777 

9782 

9786 

9791 

9795 

9800 

9805 

9809 

9814 

9818  i 

06 

9823 

9827 

9832 

9836 

9841 

9845 

9850 

9854 

9859 

9863 

07 

9868 

9872 

9877 

9881 

9886 

9890 

9894 

9899 

9903 

9908  : 

08 

9912 

9917 

9921 

9926 

9930   9934 

9939 

9943 

9948 

9952  | 

00 

9956 

9961 

9965 

9969 

9974    9978 

9983 

9987 

9991 

9996 

BIBLIOGRAPHY  AND  AUTHOR  INDEX 

Black-faced  type  refers  to  pages  in  this  treatise.    Abbreviations  are  those  used  by  Chemical 
Abstracts.     The  titles  of  articles  are  printed  in  the  original  language  whenever  convenient. 

ABEGG,  R.  Untersuchungen  iiber  Diffusion  in  Wasserigen  Salzlosungen. 
Z.  Physik.  Chem.  11,  248  (1892);  16  pp. 

ABRAMS,  A.  Viscosity  and  the  Sticking  Strength  of  Binders.  Textile 
World  J.  55,  41  (1919);  5pp. 

ABRAMS,  V.  R.,  KAVANAGH,  J.  T.,  AND  OSMOND,  C.  H.  Air  Bubble  Vis- 
cometer.  Chem.  and  Met.  Eng.  25,  665  (1921);  2  pp. 

ACHALME,  P.  Viscosity  and  Diastatic  Reactions.  Hypothesis  Concerning 
the  Nature  of  the  Diastases.  Compt.  rend.  152,  1621  (1911);  4  pp. 

ACHALME,  P.  &  BRESSON,  M.  (1)  Influence  of  Viscosity  of  the  Medium 
on  Enzyme  Reactions.  Compt.  rend.  152,  1328  (1911);  3  pp.;  (2)  Role 
of  Viscosity  in  the  Variation  of  the  Action  of  Invertase  According  to  the 
Conductivity  of  Saccharose.  Compt.  rend.  152,  1420  (1911);  3  pp. 

ADAM.  (1)  Viscositat  des  Plasma.  Berlin  klin.  Wochenschr.  (1909);  (2) 
Zur  Viscositat  des  Blutes.  Z.  klin.  Med.  68,  177,  1009  (1910);  28  pp. 

ADAMS,  F.  &  NICHOLSON,  J.  Experimental  Investigation  into  the  Flow  of 
Marble.  Phil.  Trans.  195,  363  (1901);  38  pp.  Proc.  Roy.  Soc.  London 
67,228  (1900);  6pp. 

ADAMS  PRIZE  ANNOUNCEMENT.  On  the  Criterion  of  the  Stability  &  Insta- 
bility of  the  Motion  of  a  Viscous  Fluid.  Phil.  Mag.  (5)  24,  142  (1887); 
See  Proc.  Math.  Soc.  11,  57  ( ). 

AIGNAN,  A.  Ecoulement  de  1'eau  dans  un  tuyau  cylindrique.  J.  physique 
(3)  5,  27  (1896);  2  pp. 

ALBANESE,  M.  (1)  tlber  den  Einfluss  der  Zusammensetzung  der  Ernah- 
rungsfliissigkeiten  auf  die  Thatigkeit  des  Froschherzens.  Arch,  fur 
exp.  Path.  u.  Pharm.  32,  297  (1893);  (2)  Influence  des  proprietes  phy- 
siques des  solutions  sur  le  coeur  de  grenouille.  Arch.  Ital.  de  Biol.  25, 
308  (1896);  (3)  Influenza  degli  elettroliti  sulla  Viscosita  dei  liquidi 
colloidali.  Arch.  exp.  Path.  Pharm.  (1908);  suppl.  16,  13  pp. 

ALBANESE,  V.     Arch.  Ital.  Biol.  50,  387  (1909). 

ALEKSIEJEW,  U.  AND  CERMIATASCHENSKI,  P.  A.  Zap.  imp.  russk.  techn. 
obschtsch.  30,  pt.  6-7  (1896). 

ALFORD,  L.  P.  Bearings  and  their  Lubrication.  McGraw-Hill  Book  Co. 
N.  Y.  (1911);  235  pp. 

ALLAN,  G.  Quantitative  Demonstration  Experiments  to  Show  Gaseous 
Friction.  Physik.  Z.  10,  961  (1910). 

ALLEN,  H.  6  (1)  The  Motion  of  a  Sphere  in  a  Viscous  Liquid.  Phil. 
Mag.  (5)  50,  323  (1900);  16  pp.;  (2)  Do.,  Phil.  Mag.  (5)  50,  519  (1900); 
16  pp.;  (3)  Molecular  Layers  in  Lubrication.  Proc.  Phys.  Soc.  London 
II,  32,  16  (1920);  1  p. 

347 


348  INDEX 

AMERLING,  K.  Viscosity  of  the  Blood  in  the  New  Born  and  in  Infants. 
Casopis  lek.  cesk.  (1909). 

ANDRADE,  E.  N.  DA  C.  236,  The  Viscous  Flow  in  Metals,  and  Allied 
Phenomena.  Proc.  Roy.  Soc.  London  (A)  84,  1  (1911);  12  pp.  Physik 
Z.  11,  709  (1911);  7pp. 

ANDREWS,  T.  The  Loss  in  Strength  in  Iron  and  Steel  by  Use.  Nature  65, 
418  (1896-97). 

ANON.  Vorlage  der  deutschen  Sektion  fur  die  Hauptversammlung.  Inter- 
nationalen  Kommission.  Wien.  Jan.  (1912).  Exposition  of  Osborne 
Reynold's  Theory  of  Lubrication.  Mathematical  Eng.  July  30, 
(1915).  Colloidal  Solid  Lubricants.  Do.  66,  169  (1920);  1  p.  Lubri- 
cants and  Lubrication.  Chem.  Trade  J.  67,  769  (1921);  1  pp.; 
Reduction  of  Viscosity  of  Celluloid  and  Acetone  by  Additions  of  Oxalic, 
Tartaric,  and  Citric  Acids.  Kolloid.-Z.  27,  44  (1920).  Ger.  Pat. 
276,661;  Viscosity  Determinations  in  Absolute  Units.  Eng.  105, 
655  (1918).  Comment  on  Friction  at  High  Velocities.  Proc.  Inst. 
Mech.  Eng.  660  (1883).  Comite  Deutscher  Verband  f.  die  Materiol- 
prufung  der  Technik.  Jahresber.  der  chem.  Technol.  489  (1904) ; 
5  pp.;  (1)  Report  of  Lubricants  and  Lubrication  Inquiry  Committee. 
Dept.  Sci.  &  Ind.  Research,  Advisory  Council,  (1920);  216  pp.  Cp. 
Hyde;  (2)  Memorandum  on  Cutting  Lubricants  and  Cooling  Liquids 
and  on  Skin  Diseases  Produced  by  Lubricants.  Do.  Bull.  No.  2 
(1918);  8  pp.;  (3)  Memorandum  on  Solid  Lubricants.  Bull.  No.  4 
(1920);  28  pp.;  (4)  Reports  on  Colloid  Chemistry  and  Its  General  and 
Industrial  Applications.  Cp.  Hatschek  and  also  McBain.  Do.  (1917- 
1920).  The  Determination  of  Viscosity  as  Relative  Viscosity.  Arch. 
Rubbercult.  4,  123  (1920);  14  pp.  Tentative  Test  for  the  Viscosity  of 
Lubricants.  Proc.  Am.  Soc.  Testing  Materials  I,  19,  728  (1919);  4  pp. 

APPELL,  P.  Sur  1'equation  diff6rentielle  du  mouvement  d'un  projectile 
sph6rique  pesant  dans  1'air.  Arch.  d.  Math.  u.  Phys.  (3)  5,  177  (1903); 
2pp. 

APPLEBEY,  M.  P.  16,  The  Viscosity  of  Salt  Solutions.  J.  Chem.  Soc. 
97,  2000  (1910);  26  pp.  Proc.  Chem.  Soc.  26,  266  (1910).  The 
Determination  of  Viscosity.  J.  Chem.  Soc.  103,  2167  (1913);  4  pp. 
Proc.  Chem.  Soc.  29,  361  (1913). 

ARAGO,  BABINET,  POIBERT  AND  REGNAULT  CP.  POISEUILLE.  Rapport 
fait  a  I'Acaddmie  des  Sciences  sur  un  M6moire  de  M.  le  docteur  Poi- 
seuille,  ayant  pour  titre:  Recherches  exp6rimentales  sur  le  mouvement 
des  liquides  dans  les  tubes  de  tres-petits  diametres.  Ann.  de  Chim.  1, 
50  (3);  25  pp. 

ARCHBUTT,  L.  D.  Tests  for  Curcas  Oil.  J.  Soc.  Chem.  Ind.  17, 1009  (1898) ; 
Lubricators  and  Lubrication.  Chem.  Age  (London)  4,  280  (1920);  1  p. 

ARCHBUTT,  L.  &  DEELEY,  R.  Lubrication  and  Lubricants,  a  Treatise  on  the 
Theory  and  Practice  of  Lubrication  and  on  the  Nature,  Properties 
and  Testing  of  Lubricants.  London,  Griffin,  3d  Ed.  (1912);  589  pp. 
Cp.  Deeley. 


INDEX  349 

ARISZ,  L.  289,  Kolloidchem.  Beihefte  7,  1  (1915);  90  pp.  Sol  und  Gelzus- 
tand  von  Gelatinelosungen. 

ARNDT,  K.  6,  (1)  Zahigkeitmessungen  bei  hohen  Temperaturen.  Z. 
Elektrochem.  13,  578  (1907);  (2)  Zahigkeit  und  Leitfahigkeit.  Z. 
Elektrochem.  13,  809  (1907);  (3)  Messung  der  Zahigkeit.  Z.  Apparat- 
kunde  3,  473. 

ARNDT,  K.-U.  GESSLER  A.     Z.  Elektrochem.  13,  580  (1907). 

ARNDT,  K.  and  SCHIPF,  P.     Kolloidchem.  Beihefte  6,  201  (1914). 

ARNOLD,  H.  D.  Phil.  Mag.  (6)  22,  755  (1911);  21  pp.  Limitations  imposed 
by  Slip  and  Inertia  Terms  upon  Stokes'  Law  for  the  Motion  of  Spheres 
through  Liquids. 

ARONHEIM.  tlber  den  Einfluss  der  Salze  auf  die  Stromungsgeschwindigkeit 
des  Blutes.  Diss.  Gottingen  (1868).  Cf.  Hoppe  Seyler's  Med.  Chem. 
Unterssuchungen  Berlin  265  (1867). 

ARONS,  DAMMER.     Chem.  Tech.  1,  771.     Plasticity. 

ARPI,  R.  Experimental  Determination  of  Viscosity  and  Density  of  Certain 
Molten  Metals  and  Alloys.  Intern.  Zeitschr.  Metallog.  6,  142  (1913); 
26pp. 

ARRHENIUS,  S.  3,  7,  (1)  Contributions  to  our  Knowledge  of  the  Action  of 
Fluidity  on  the  Conductivity  of  Electrolytes.  Brit.  Assoc.  Rep.  344 
1886);  4  pp.;  (2)  tlber  die  innere  Reibung  verdunnter  Wasseriger 
Losungen.  Z.  physik.  Chem.  1,  285  (1887);  14  pp.;  (3)  Electrolytic 
Dissociation  versus  Hydration.  Phil.  Mag.  (5)  28,  39  (1889);  9  pp.; 
(4)  Z.  P.  C.  9,  487  (1892);  25  pp.  tlber  die  A'nderung  des  elektrischen 
Leitungsvermogens  einer  Losung  durch  Zusatz  von  kleinen  Mengen 
eines  Nichtleiters;  (5)  Viscosity  and  Hydration  of  Colloid  Solutions. 
Medd.  fram.  K.  Vetenskapsakad.  Nobelinst.  3,  (1916);  (6)  The  Viscosity 
of  Pure  Liquids.  Do.  3,  No.  20  (1912);  40  pp.;  (7)  The  Viscosity  of 
Solutions.  Biochem.  J.  11,  112  (1917);  22  pp. 

ASHLEY,  H.  E.  (1)  The  Colloid  Matter  in  Clay,  65  pp.  Trans.  Am.  Ceram. 
Soc.  11,  530  (1909);  66  pp.;  (2)  The  Colloidal  Matter  in  Clay  and  its 
measurement.  Bull.  388,  U.  S.  Geol.  Surv.  66  pp.  U.  S.  Bureau  of 
Standards.  Bull.  23,  (1913). 

ATEN,  A.  Electrical  Conductivity  in  Mixtures  of  Metals  and  their  Salts. 
Z.  physik.  Chem.  66,  641  (1909);  31  pp. 

ATKINS  &  WALLACE.  J.  Chem.  Soc.  103,  1461  (1913).  Molecular  con- 
dition of  Mixed  Liquids  I.  Mixtures  of  Lower  Aliphatic  Alcohols 
with  Water. 

ATTERBERG,  A.  The  plasticity  of  clay.  Intern.  Mitt.  Bodenkunde  1, 
4  (1910);  33  pp.  Cp.  Tonind.  Ztg.  36,  1460;  (2)  Barium  Sulphate  a 
Plastic  Substance.  Z.  angew.  Chem.  24,  928  (1910) ;  1  p. ;  (3)  The  Plas- 
ticity of  Barium  Sulphates.  Z.  angew.  Chem.  24,  2355.  Cp.  Ehien- 
berg;  (4)  Internat.  Mitt.  Bodenk.  3,  291  (1914);  39  pp.;  (5)  The  Plas- 
ticity and  Coherence  of  Clays  and  Loams.  Chem.  Ztg.  34,  369;  2  pp. 
379  (1910);  1  p.;  (6)  What  constituents  give  to  clay  its  plasticity  and 
firmness?  Kgl.  Landtbruks  Akad.  Handlingar  och  Tidskrift  413 
(1913);  32pp. 


,350  INDEX 

VAN  AUBEL,  EDM.     Compt.  rend.  173,  384  (1921);  4  pp.     Influence  de  la 

temperature  sur  la  viscosite  des  liquides  normaux. 
AUERBACH,    F.     (1)    Magnetische    Untersuchung.     Wied.    Ann.    14,    30« 

(1881);  (2)  Plasticitat  und  Sprodigkeit.     Wied.  Ann.  46,  277  (1892); 

15  pp. 
AUGENHEISTER.     Bcitrage    zur    Kenntniss    der    Elasticitat    der    Metalle. 

Diss.  Berlin  (1902). 
AUSTIN,   L.     Experimentaluntersuchungen   iiber   die   elastiche   Langs-und 

Torsionsnachwirkung  in   Metallen.     Wied.   Ann.   60,  659   (1893);   19 

pp.;  Cp.  Nature  49,  239  (1894). 
AXELROD,    S.     Gummizeitung    19,    1053    (1905);     Gummizeitung    20,  105 

(1905);  Gummizeitung  23,  810  (1910).     The  Viscosity  of  Caoutchouc 

Solutions.     Cp.  Gummizeitung  19,  1053  &  20,  105. 
AXER,  J.     A  Viscometer  Consisting  of  a  Vertical  Cylinder  Filled  with  the 

Material  into   Which  Another  Perforated  Cylinder  Dips.     Ger.  pat. 

267,  917  (1913). 
AYRTON,  W.  &  PERRY  J.     On  the  Viscosity  of  Dielectrics.     Proc.  Roy.  Soc. 

27,238  (1878);  7  pp. 

BACHMANN.  (1)  Die  klinische  Verwertung  der  Viscositatsbestimmung. 
Deutsch.  Arch.  klin.  Med.  94  (1908);  (2)  Die  Viscositat  des  Blutes  und 
ihre  diagnostische  Bedeutung.  Med.  Klinik  36  (1909). 

BAPFANI  &  LUCIANI.  Physico-chemical  Investigation  of  the  Blood  of  the 
Mother  &  the  Fetus  with  Especial  Reference  to  the  Viscosity.  Atti. 
soc.  ital.  ostetricia  (1908);  Arch.  ital.  biol.  61,  246  (1900);  7  pp. 

BAHEUX,  CH.  Comparison  of  Viscometers.  Mat.  grasses  6,  3231  (1911); 
3  pp.  (Report  on  Barbey  and  Engler  types.) 

BAILY,  F.  (1)  On  the  Correction  of  a  Pendulum  for  the  Reduction  to  a 
Vacuum  together  with  Remarks  on  some  Anomalies  observed  in  Pendu- 
lum Experiments.  Phil.  Trans.  122,  399  (1832);  98  pp.;  (2)  Do.  Phil. 
Mag.  (3)  1,379  (1832);  3pp. 

BAKER,  F.  (1)  Viscosity  of  Ether-Alcohol  Mixtures.  J.  Chem.  Soc. 
101, 1409  (1912);  7  pp.;  (2)  The  Viscosity  of  Cellulose  Nitrate  Solutions. 
J.  Chem.  Soc.  103,  1653  (1913);  22  pp. 

BALDUS,  A.  A  Viscosity  Meter.  Brit.  22,961,  Oct.  4  (1910).  (Air  bubble 
method.) 

BANCELIN,  M.  (1)  The  Viscosity  of  Emulsions.  Compt.  rend.  162,  1382 
(1911);  2  pp.;  (2)  The  Viscosity  of  Suspensions  &  the  Determination 
of  Avogadro's  Number.  Z.  Chem.  Ind.  Kolloide  9,  154  (1912);  2  pp. 

BANCROFT,  W.  D.  213,  Applied  Colloid  Chemistry.  General  Theory; 
345  pp.  McGraw-Hill  Book  Co. 

BARBEY.     324,  See  Nicolardot,  Baheux. 

BARNES,  J.  (1)  On  the  Relation  of  the  Viscosities  of  Mixtures  of  Solutions 
of  Certain  Salts  to  their  State  of  lonization.  Proc.  and  Trans.  Nova 
Scotia  Inst.  of  Sci.  10,  113  (1899);  26  pp.;  (2)  Do.,  Elektrochem. 
Z.  7,  134  (1900);  6pp.;  (3)  Do.,  Chem.  News  86,  4,  30,  40  (1902); 
6pp. 


INDEX  351 

BARNETT,  R.  On  the  Viscosity  of  Water  as  Determined  by  Mr.  J.  B. 
Hannay  by  means  of  his  Microrheometer.  Proc.  Roy.  Soc.  London 
66,259  (1894);  3pp. 

BARR,  G.  and  BIRCUMSHAW,  L.  L.  The  Viscosity  of  Some  Cellulose  Acetate 
Solutions.  Advisory  Comm.  for  Aeronautics.  Reports  and  Memo- 
randa, No.  663  (1919);  6  pp. 

BARTHELEMY,  H.  The  Measurement  of  Viscosity  and  the  Cellulose  Ester 
Industries.  Caoucthouc  and  Guttapercha  10,  7202  (1910);  11  pp. 

BARTOLI,  A.  Bull.  Mensile  Accad.  Giornale  di  Sci.  Natur.  Catania  26,  4 
(1892). 

BARTOLI,  A.  &  STRACCIATI,  E.  (1)  Le  Proprieta  fisiche  degl'  idrocarburi 
Cn  H2«  +  2  dei  petrolii  di  Pensilvania.  Cim.  (3)  18,  195  (1885); 
24  pp. ;  (2)  Sur  les  proprictes  physiques  des  hydrocarbures  CnEUn  +  2 
des  petroles  d'Amerique.  Ann.  chim.  phys.  (6)  7,  375  (1886);  9  pp. 

BARUS,  C.  77,  212,  235,  246,  286  (1)  The  Electrical  and  Magnetic  Proper- 
ties of  Iron,  Bull.  #14,  U.  S.  Geol.  Survey  (1888);  (2)  Note  on  the  Vis- 
cosity of  Gases  at  high  Temperatures  and  on  the  pyrometric  Use  of  the 
Principle  of  Viscosity.  Am.  J.  Sci.  (3)  35,  407  (1888);  68  pp.;  (3) 
Maxwell's  Theory  of  the  Viscosity  of  Solids  and  its  Physical  Verification. 
Phil.  Mag.  (5)  26,  183  (1888);  (4)  Die  Zahigkeit  der  Case  bei  hohen 
Temperaturen.  Wied.  Ann.  36,  358  (1889);  41  pp.;  (5)  The  Viscous 
Effect  of  Strains  Mechanically  Applied,  as  Interpreted  by  Maxwell's 
Theory.  Phil.  Mag.  (5)  27,  155  (1889);  23  pp.;  (6)  On  Thermo- 
Electric  Measurements  of  High  Temperatures.  Bull.  #54,  U.  S.  Geol. 
Survey  (1889);  (7)  The  change  of  the  Order  of  Absolute  Viscosity 
Encountered  on  Passing  from  Fluid  to  Solid.  Phil.  Mag.  (5)  29, 
337  (1890);  19  pp.;  (8)  The  Viscosity  of  Solids.  Bull.  #73,  U.  S.  Geol. 
Survey  (1891);  139  pp.;  (9)  The  Thermal  Variation  of  Viscosity  and  of 
Electrolytic  Resistance.  Am.  J.  Sci.  (3)  44,  255  (1892);  1  p.;  (10) 
Isothermals,  Isopiestics,  and  Isometrics  relative  to  Viscosity.  Am.  J. 
Sci.  (3)  45,  87  (1893);  10  pp.;  (11)  Note  on  the  Dependence  of  Viscosity 
on  Pressure  and  Temperature.  Proc.  Am.  Acad.  N.  S.  19,  13  (1893); 
6  pp.;  (12)  Mechanism  of  Solid  Viscosity.  Bull.  #94,  U.  S.  Geol. 
Survey;  (13)  Maxwell's  Theory  of  Viscosity.  Am.  J.  Sci.  36,  178 
(1888). 

BARUS,  C.  &  STROUHAL.  (1)  The  Viscosity  of  Steel  and  its  Relation  to 
Temper.  Am.  J.  Sci.;  (3)  32,  444  (1886);  23  pp.;  (2)  Do.,  Concluded. 
Am.  J.  Sci.  (3)  33,  20  (1887);  16  pp. 

BARY,  P.  The  Viscosity  of  Colloidal  Solutions.  Compt.  rend.  170,  1388 
(1920);  2  pp. 

BASCH,  K.  Viscometric  Studies  of  Human  Milk.  Wien.  Klin.  Wochschr. 
24,  1592  (1910);  3  pp. 

BASSETT,  A.  B.  Treatise  of  Hydrodynamics.  2  vols.  Deighton,  Bell  & 
Co.,  Cambridge  (1888);  264  +  328  pp.;  (2)  Stability  &  Instability  of 
Viscous  Liquids.  Proc.  Roy.  Soc.  52,  273  (1893);  3  pp. 

BATEMAN,  E.  Relation  between  Viscosity  and  Penetrance  of  Creosote 
into  Wood.  Chem.  Met.  Eng.  22,  359 '(1920);  2  pp. 


352  INDEX 

BATSCHINSKI,  A.  92,  132,  134,  142,  et  seq.  148,  162,  161,  166  et  scq. 
(1)  tlber  das  Gesetz  der  Veranderlichkeit  der  Viskositat  des  Queck- 
silbers  mit  der  Temperatur.  Ann.  d.  phys.  Klasse  d.  Kais.  Gesellsch. 
v.  Freunden  e.  Naturw.  zu  Moskau  10,  8  (1900);  (2)  Beziehung  zwis- 
chen  der  Viskositat  und  der  chemischen  Konstitution  der  Fliissig- 
keiten.  Sitzungsber.  d.  Kaiserl.  Ges.  d.  Naturf  zu  Moskau  (1900); 
100  pp.  Cp.  p.  1  (1901);  p.  265  (1902);  Chem.  Zentr.  2,  450  (1901); 
Do.  2,  180  (1902);  (3)  t)ber  die  Beziehung  zwischen  dem  Viscositats- 
parameter  und  anderen  physikalischen  Konstanten.  Z.  physik.  Chem. 
37,  214  (1901);  2  pp.;  (4)  Untersuchungen  tiber  die innere  Reibung  der 
Fliissigkeiten.  Annales  de  la  Societe1  d'encouragement  des  sciences 
experimentales  et  de  leurs  applications  du  nom  de  Christophe  Leden- 
zoff.  Supplement  3  (1913);  70  pp.;  (5)  Viscosity  Law  for  Liquids. 
Physik  Z.  13,  1157  (1913);  (6)  Inner  Friction  of  Liquids.  Z.  physik. 
Chem.  84,  643  (1913);  64  pp.;  (7)  Association  of  Liquids  Z.  physik. 
Chem.  82,  86  (1913);  6  pp. 

DES  BAUCELS,  LARGUIER.  Investigations  of  the  Physical  Changes  in  Gela- 
tine in  the  Presence  of  Electrolytes  and  Non-Electrolytes.  Compt. 
rend.  146,  290  (1905). 

BAUME,  G.  and  VIGNERON.  New  Apparatus  for  Measuring  Viscosities  and 
Fluidities.  Ann.  chim.  anal.  chim.  appl.  1,  379  (1915);  5  pp. 

BAYLISS,  W.  M.  286,  (1)  The  kinetics  of  tryptic  action.  Arch.  d.  Sciences 
Biologiques  11,  261  (1904);  35  pp.;  (2)  Causes  of  the  Rise  in  Electrical 
Conductivity  under  the  Action  of  Trypsin.  J.  of  Physiol.  36,  221 
(1907);  31  pp.;  (3)  The  Nature  of  Enzyme  Action.  Longmans  Green 
&Co.  (1914);  179pp. 

BAZIN.  Experiences  nouvelles  sur  la  distribution  des  vitesses  dans  les 
tuyaux.  Compt.  rend.  122,  1250  (1896);  3  pp. 

BEADLE,  C.  &  STEVENS,  H.  P.  The  Viscosity  of  Rubber  Solutions.  India 
Rubber  J.  46,  1081  (1913);  1  p. 

BECK,  C.  &  HIRSCH,  C.  (1)  Die  Viskositat  des  Blutes.  Arch.  f.  exp.  Path, 
u.  Pharm.  64,  54  (1905). 

BECK,  K.  (1)  Beitrage  zur  Bestimmung  der  Relativen  inneren  Reibung  von 
Flussigkeiten.  Habilitationschrift,  Leipzig  (1904);  (2)  Beitrage  zur 
Bestimmung  der  Relativen  inneren  Reibung  von  Flussigkeiten,  im 
besondefn  des  menschlichen  Blutes.  Z.  physik.  Chem.  48,  641  (1904); 
40  pp.;  (3)  The  Influence  of  the  Red  Corpuscles  on  the  Internal  Friction 
of  the  Blood.  Kolloid-Z.  26,  109  (1919);  1  p. 

BECK,  K.  &  EBBINGHOUSE,  K.  Beitrage  zur  Bestimmung  der  inneren 
Reibung.  Z.  physik.  Chem.  68,  409  (1899);  16  pp. 

BECKER,  A.  t)ber  die  innere  Reibung  und  Dichte  der  Bunsenflamme. 
Ann.  Physik.  (4)  24,  823  (1907);  39  pp. 

BECKER,  G.  F.     Strain  &  Rupture  in  Rocks.     Geol.  Surv.  1897-1902. 

BELL,  J.  &  CAMERON,  F.  189,  The  Flow  of  Liquids  through  Capillary 
Spaces.  J.  Phys.  Chem.  10,  658  (1906);  17  pp. 

VAN  BEMMELEN.  Z.  anorg.  Chem.  6,  466;  13,  233;  18,  14,  98;  20,  185;  22, 
313. 


INDEX  353 

32. 

BENCE,  J.  (1)  Klinische  Untersuchungen  iiber  die  Viscositat  des  Blutes 
bei  Storungen  der  CO»  Ausscheidungen.  Deutsch.  med.  Wochschr. 
#15  (1905);  (2)  Klinische  Untersuchungen  iiber  die  Viscositat  des 
Blutes.  Z.  klin.  Med.  27,  (1907);  (Abt.  fur  innere  Med.).  Do.  58, 
(1909). 

BENTON,  A.  F.  The  End  Correction  in  the  Determination  of  Gas  Viscosity 
by  the  Capillary-Tube  Method.  Phys.  Rev.  14,  403  (1919);  6  pp. 

BERL  and  KLAYE.  (The  viscosity  of  oxy-  and  hydro-cellulose  nitrates.) 
Z.  Schiess.  Sprengstoffwesen  2,  381  (1907). 

BERLAND,  E.  and  CHENVIER,  A.  Lubrication  and  Viscosity  of  Liquids. 
Memoires  de  la  Societe"  de  sciences  phys.  3,  405  (?)  (1887). 

BERNDORF,  H.  The  Determination  of  Viscosity  of  Transverse  Waves  in 
the  Outermost  Earthcrust.  Physik.  Z.  13,  83  (1912);  1  p. 

BERNOUILLI,  J.  Theoria  nova  de  motu  aquarum  per  canales  quocunque 
fluentes.  Acad.  St.  Petersbourg  (1726). 

BERNSTEIN,  G.  Studies  on  the  Vulcanization  of  Rubber.  Kolloid-Z.  11, 
185  (1912). 

BERSON,  G.  &  BOUASSE,  H.  Sur  1'clasticite  de  torsion  d'un  fil  oscillant. 
Compt.  rend.  119,  48  (1894);  2  pp. 

BESSEL.     Berlin  Acad.  (1826). 

BESTELMEYER,  A.  (1)  Die  innere  Reibung  des  Stickstoffs  bei  tiefen  Tempera- 
turen.  Diss.  Munchen  (1903);  59  pp.;  (2)  Bemerkung  zu  der 
Abhandlung  des  Herrn  Markowski  liber  die  inneren  Reibung  von  Sauer- 
stoff,  Wasserstoff,  chemischen  und  atmospharischen  Stickstoff  und 
ihre  A'nderung  mit  der  Temperatur.  Ann.  Physik.  (4)  15,  423  (1904); 
2pp. 

BIEL,  R.  Uber  den  Druckhohenverlust  bei  der  Fortleitung  tropbarer  und 
gasformiger  Fliissigkeiten.  Mitt.  Forschungsarbeiten  Verein  deutscher 
Ingenieure,  Heft  44.  Springer  Berlin  (1907),  abst.  Zeitschr.  d.  Ver. 
deutsch.  Ing.  1035,  1065  (1908);  59pp. 

BILTZ,  W.  &  VON  VEGESACK,  A.  &  Steiner,  H.  205,  Uber  den  osmotischen 
Druck  der  Kolloide.  II..  Der  osmotische  Druck  einiger  Farbstoff- 
losungen.  Section  5.  Uber  die  Zahigkeit  von  Nachtblaulosungen. 
Z.  physik.  Chem.  73,  500  (1910). 

BINGHAM,  E.  C.  126,  142,  271,  289,  (1)  The  Conductivity  and  Viscosity  of 
Solutions  of  Certain  Salts  in  Mixtures  of  Acetone  with  Methyl  Alcohol, 
Ethyl  Alcohol  and  Water.  Diss.  Johns  Hopkins  (1905);  78  pp.  Cp. 
Jones  &  Bingham;  (2)  Viscosity  and  Fluidity.  Am.  Chem.  J.  35,  195 
(1906);  23  pp.;  (3)  Viscosity  and  Fluidity.  Am.  Chem.  J.  40,  (1908); 
4  pp.;  (4)  Viskositat  und  Fluiditat.  Z.  physik.  Chem.  66,  238  (1909); 

17  pp.;  (5)  Viscosity  &  Fluidity.     Am.  Chem.  J.  43,  287  (1910);  23  pp.; 
(6)  Viscosity  and  Fluidity  of  Matter  in  the  Three  States  of  Aggregation, 
and  the  Molecular  Weight  of  Solids.     Am.  Chem.  J.  45,  264  (1911); 

18  pp.;   (7)   Fluidity  and  Vapor  Pressure.     Am.   Chem.   J.  47,   185 
(1912);  12  pp.;  (8)  Viscosity  &  Fluidity.     A  Summary  of  Results.     I. 
Phys.  Rev.  35,  (1912) ;  26  pp. ;  (9)  Viscosity  &  Fluidity.     A  Summary  of 

23 


354  INDEX 

Results.  II.  Phys.  Rev.  (2)  1,  96  (1913);  27  pp.;  (10)  A  Criticism  of 
Some  Recent  Viscosity  Investigations.  J.  Chem.  Soc.  103,  959  (1913); 
6  pp.;  Proc.  Chem.  Soc.  29,  113  (1913);  (11)  The  Viscosity  of  Binary 
Mixtures.  J.  Phys.  Chem.  18,  157  (1914);  8  pp.;  (12)  A  New  Vis- 
cometer  for  General  Scientific  &  Technical  Purposes.  J.  Ind.  Eng. 
Chem.  6,  233  (1914);  8  pp.;  (13)  Fluidity  as  a  Function  of  Volume, 
Temperature  and  Pressure.  The  Equation  of  State  and  the  Two 
Kinds  of  Viscous  Resistance.  The  So-called  "Slipping"  in  Gases. 
J.  Am.  Chem.  Soc.  36,  1393  (1914);  16  pp.;  (14)  A  review  of  Dunstan 
and  Thole's  "Viscosity  of  Liquids."  J.  Am.  Soc.  36,  1320  (1914); 
2  pp.;  (15)  Plastic  Flow.  J.  Wash.  Acad.  Sci.  6,  177  (1916);  3  pp.; 
(16)  An  Investigation  of  the  Laws  of  Plastic  Flow.  Bull.  U.  S.  Bur.  of 
Standards  13,  309  (1916);  43  pp.;  (17)  The  Variable  Pressure  Method 
for  the  Measurement  of  Viscosity.  Proc.  Am.  Soc.  Testing  Materials 
18,  Pt.  II,  373  (1918);  11  pp.;  (18)  Cutting  Fluids.  Tech.  Paper 
204,  U.  S.  Bur.  of  Standards  16,  35  (1922)  41  pp. 

BINGHAM,  E.  C.  and  DURHAM,  T.  C.  64,  201,  208,  215,  The  Viscosity  and 
Fluidity  of  Suspensions  of  Finely-Divided  Solids  in  Liquids.  Am. 
Chem.  J.  46,  278  (1911);  20  pp. 

BINGHAM,  E.  C.  and  GREEN,  H.  220,  et  seq.,  Paint,  a  Plastic  Material. and 
not  a  Viscous  Liquid;  The  Measurement  of  Its  Mobility  and  Yield 
Value.  Proc.  Am.  Soc.  Testing  Materials.  II,  19,  640  (1919);  36  pp. 
Cp.  Green. 

BINGHAM,  E.  C.  &  Miss  HARRISON,  J.  121,  Viskositat  und  Fluiditat.  Z. 
physik.  Chem.  66,  1  (1909);  32  pp. 

BINGHAM,  E.  C.  and  JACKSON,  R.  F.  Standard  Substances  for  the  Cali- 
bration of  Viscometers.  Bull.  U.  S.  Bur.  of  Standards  14,  No.  298, 
59  (1917);  28  pp.  J.  Wash.  Acad.  Sci.  7,  53  (1917);  2  pp. 

BINGHAM,  E.  C.  and  SARVER,  L.  Fluidities  and  Specific  Volumes  of  Benzyl 
Benzoate  and  Benzene.  J.  Am.  Chem.  Soc.  42,  2011  (1920);  11  pp. 

BINGHAM,  E.  C.,  SCHLESINGER,  H.  I.  and  COLEMAN,  A.  B.  298,  Some 
Sources  of  Error  in  Viscosity  Measurement.  J.  Am.  Chem.  Soc. 
38,  27  (1916);  15  pp. 

BINGHAM,  E.  C.,  VAN  KLOOSTER,  H.  S.  and  KLEINSPEHN,  W.  G.  The 
Fluidities  and -Volumes  of  Some  Nitrogenous  Organic  Compounds. 
J.  Phys.  Chem.  24,  1  (1920);  21  pp. 

BINGHAM,  E.  C.  &  WHITE,  G.  F.  6,  21,  28,  97,  (1)  Viscosity  and  Fluidity 
of  Emulsions,  Crystallin  Liquids  and  Colloidal  Solutions.  J.  Am.  Soc. 
33,  1257  (1911);  11  pp.;  (2)  Fluiditat  und  die  Hydrattheorie.  I.  Die 
Viskositat  von  Wasser.  Z.  physik.  Chem.  80,  670  (1912);  17  pp. 

BINGHAM,  E.  C.,  WHITE,  G.  F.,  THOMAS,  A.  &  CADWELL,  J.-L.  142,  169, 
178,  Fluidity  and  the  Hydrate  Theory.  II.  Z.  physik.  Chem.  83, 
641  (1913);  32  pp. 

BLANCHARD,  A.  The  Viscosity  of  Solutions  in  Relation  to  the  Constitution 
of  Dissolved  Substances.  J.  Am.  Chem.  Soc.  26,  1315  (1904);  24  pp. 
(Ionic  migration  velocity.) 

BLANCHARD,  A.  &  PUSHES,  H.  B.     The  Viscosity  of  Solutions  of  the  Metal 


INDEX  355 

Ammonia  Salts.     J.  Am.  Chem.  Soc.  34,  28  (1912);  4  pp.     Cp.  J.  Am. 

Chem.  Soc.  26,  1315. 
BLANCHARD,  A.  &  STEWART,  M.     The  Viscosity  of  Solutions  of  Metallic 

Salts;  Its  Bearing  upon  the  Nature  of  the  Compound  between  Solvent 

and  Solute.     Science  (N.  S.)  18,  98  (1903);  1  p. 
BLASIUS,   H.     Das  Ahnlichkeitsgesetz  bei  Reibungsvorgangen  in  Flussig- 

keiten.     Mitt.  Forschungsarbeiten,  Verein  deutscher  Ingenieure.    Julius 

Springer.     Heft.  131  (1913);  40  pp. 
BLEININGER,   A.   V.     (1)   The  Effect  of  Preliminary  Heating  Treatment 

Upon  the  Drying  of  Clays.     Technologic  Paper,  Bur.  of  Standards  #1 

(1911);    (2)   The  Viscosity  of  Clay  Slips.     Trans.   Am.   Ceram.   Soc. 

10,  389;  (3)  The  Effect  of  Electrolytes  upon  Clay  in  the  Plastic  State. 

Orig.  Comm.  8th  Intern.  C.  Appl.  Chem.     Cp.  Univ.  111.  Bull.  6,  #25 

(1909). 
BLEININGER,  A.  V.  &  BROWN,  G.  H.     (1)  Testing  of  Clay  Refractories 

with  Special  Reference  to  their  Load  Carrying  Capacity  at  Furnace 

Temperatures.     Tech.  Paper  of  the  U.  S.  Bur.  of  Standards  #7;  (2) 

Note  on  the  V.  of  Clay  Slips.     Trans.  Am.  Ceram.  Soc.  11,  596  (1909); 

9  pp.;  (3)  The  Effect  of  Prelim.  Heat  Treatment  upon  Clays.     U.  S. 

Bur.  Stand.  Bui.  1.     Trans.  Am.  Ceram.  Soc.  11,  392  (1909);  15  pp. 
BLEININGER,  A.  V.,  CLARK,  H.  H.     Note  on  the  Viscosity  of  Clay  Slips  as 

Determined  by  the  Clark  Apparatus.     Trans.  Am.  Ceram.  Soc.  12, 

383  (1910);  9pp. 
BLEININGER,  A.  V.  &  FULTON,  C.  E.     The  effect  of  Acids  and  Alkalies  upon 

Clay  in  the  Plastic  State.     Am.  Ceram.  Soc.  14,  827  (1912). 
BLEININGER,  A.  V.  &  Ross,  D.  W.     (1)  The  Flow  of  Clay  under  Pressure. 

.     Trans.  Am.  Ceram.  Soc.  16,  392  (1914);  9  pp. 
BLEININGER,  A.  V.  and  TECTOR,  P.     The  Viscosity  of  Porcelain  Bodies. 

Trans.   Am.    Ceram.  Soc.  16,  328  (1913);  9  pp.     Also  Tech.  Paper, 

U.  S.  Bur.  Standards  #30. 
BLUNSCHY,  F.     Beitrage  zur  Lehre  der  Viscositat  des  Blutes.     Diss.  Zurich 

(1908);  43  pp. 
BOGUE,  R.  H.     Properties  and  Constitution  of  Glues  and  Gelatins.     II. 

Chem.  Met.  Eng.  23,  61  (1920);  6  pp.;  J.  Am.  Chem.  Soc.  43,  1764 

(1921);  10  pp.     The  Viscosity  of  Gelatin  Sols. 
BOLLE,  B.     Beitrag  zur  Kenntniss  der  Viscositat  des  Blates,  des  Serums  u. 

des  Plasmas.     Diss.  Berlin  (1909);  29  pp. 
BOLTZMANN,  L.     237,   (1)  Zur  Theorie  der  elastische  Nachwirkung   (2A) 

Wien.  Sitzungsber.  70,  275  (1875);  31  pp.;  (2)  Zur  Theorie  der  elastis- 

chen    Nachwirkung.     Pogg.    Ann.    Erganzungsband    7,    624    (1876); 

31  pp.;  (3)  Uber  einige  Probleme  der  Theorie  der  elastischen  Nach- 
wirkung und  liber  eine  neue  Methode,  Schwingungen  mittels  Spiegel- 

ablesung  zu  beobachten,  ohne  den  Schwingenden  Korper  mit    einen 

Spiegel  von  erheblicher  Masse  zu  belasten.     Wien.  Situngsber.  (2A) 

76,  815  (1878);  28  pp.;  (4)  Zur  Theorie  der  elastische  Nachwirkung. 

Wied.  Ann.  5,  430  (1878);  3  pp.;   (5)   Zur  Theorie  der  Gasreibung. 

Wien.  Sitzungsber.   (2A)  81,  117   (1880);  42  pp.;   (6)   Do.,  Part  II, 


356  INDEX 

Wien.  Sitzungsber.  (2A)  84,  (1881);  95  pp.;  (7)  Do.,  Part  III.  Wicn. 
Sitzungsber.  (2A)  84,  1230  (1881);  34  pp.;  (8)  Vorlesungen  iiber 
Gastheorie.  Earth,  Leipzig  2  vols.  (1896);  (9)  Geschichte  unserer 
Kenntniss  der  inneren  Reibung  und  Warmeleitung  in  verdunnten 
Gasen.  Physik.  Z.  1,  213  (1900);  1  p. 

BOND,  W.  N.  The  Properties  of  Plastic  Crystals  of  Ammonium  Nitrate. 
Phil.  Mag.  41,  1  (1921);  21  pp. 

BORELLI  L.  &  DATTA.  (1)  Viscometry  of  the  Urine.  Riv.  crit.  clin.  Med. 
11,  289  (1910);  7  pp.;  (2)  Saggi  di  viscosimetria  clinica.  La  clinica 
medica  italiana  45,  149  (1905);  (3)  Saggi  di  viscosimetria  clinica. 
Nota  II.  Viscometria  degli  essudati  e  trasudati.  Riv.  crit.  clin.  Med. 
7,  181  (1906). 

BORN,  M.  The  Mobility  of  the  Electrolytic  Ions.  Z.  Elektrochem.  26, 
401  (1920);  3  pp. 

BOSCOVICH,  R.  J.  1,  Opera  pertinentia  ad  opticam  et  Astronomician. 
5  vols.  (1785)  Bassani  vol.  5  opusculum  III. 

BOSE,  E.  96,  100,  102,  209,  210,  (1)  Uber  die  Viskositatsanomalien  von 
Emulsionen  und  von  anistropen  Fliissigkeiten.  Physik.  Z.  9,  707 
(1908);  1  p.;  (2)  Viskositatsanomalien  anistroper  Flussigkeiten  in 
hydraulischen  Stromungzustanden.  Physik.  7.  10,  32  (1909);  5  pp. 
Cp.  Willers. 

BOSE,  E.  &  BOSE,  M.  The  Viscosity  of  Liquids  in  the  Condition  of  Turbu- 
lent Flow.  Physik.  Z.  12,  126  (1910);  10  pp. 

BOSE,  E.  &  CONRAT,  F.  The  Viscosity  at  the  Clarifying  Point  of  So-called 
Crystalline  Liquids.  Physik.  Z.  9,  169  (1908);  5  pp. 

BOSE,  E.  &  RAUERT,  D.  97,  Experimental  Study  of  the  Viscosity  of  Liquids 
in  the  Condition  of  Turbulent  Flow.  Physik.  Z.  10,  406  (1909);  3  pp. 

BOSSUT.  1,  18,  (1)  Traite"  61ementaire  d'hydrodynamique.  Paris  (1775); 
(2)  Nouvelles  exp6riences  sur  la  resistance  des  fluides,  Paris  (1777). 

BOTAZZI,  F.  (1)  L'Orsi,  giornale  di  chimica,  formacia,  ecc.  Firenze  20, 
253,  289  (1897).  Chem.  Zentr.  1,  83  (1898);  (2)  Recherches  sur  la 
viscosit6  de  quelques  liquides  organiques  et  de  quelques  solutions 
aqueuses  de  substances  prot&ques.  Arch.  ital.  de  Biologie  29,  401 
(1898);  Naturw.  Rundsch  14,  47  (1899);  (3)  Ricerche  sull'  attrito 
interno  (viscosita)  di  alcune  liquidi  organic!  e  di  alcune  soluzioni  acquose 
di  sostanze  proteiche.  Principi  di  Fisiologia  I  Chimica-Fisica  Societa 
editrice  libraria  316  (1906);  (4)  Some  colloidal  properties  of  hemo- 
globin. Atti.  Acad.  Lincei  22,  II,  263  (1913). 

BOTAZZI,  F.  &  D' Agostino,  E.  Viscosity  and  Surface  Tension  of  Suspensions 
and  Solutions  of  Muscular  Proteins  Under  the  Influence  of  Acids  and 
Alkalies.  Atti.  Accad.  Lincei  22,  II,  183  (1913);  9  pp. 

BOTAZZI,  F.  &  D'ERRICO,  G.  207,  208,  Pfliigers  Arch.  f.  Physiol.  115,  359 
Biochem.  Z.  7,  (1908). 

BOTAZZI,  F.  &  JAPPELI.  Viscosity  of  blood  serum  of  cattle.  Rend.  Line. 
(5)  17,  [2]  49  (1908),  Cp.  Japelli. 

BOTAZZI,  F.  and  VICTOROW,  C.     Surface  Tension,  Viscosity,  and  Appear- 


INDEX  357 

ance  of  Dialysed  Marseilles  Soap  of  Unknown  Composition,  With  or 
Without  Addition  of  Alkali.  Rend.  R.  Ac.  Line.  (5)  19,  I,  659  (1910). 

BOTTOMLEY,  L.  On  the  Secular  Experiments  in  Glasgow  on  the  Elasticity 
of  Wires.  Rep.  Brit.  Assoc.  537  (1886);  1  p. 

BOUSFIELD,  W.  R.  196,  Z.  physik.  Chem.  53,  303  (1905).  Ionic  Size  in 
Relation  to  the  Physical  Properties  of  Silver  Solutions.  Phil.  Trans. 
A.  206,  129  (1906);  13  pp.;  (2)  Ionic  Size  in  Relation  to  Viscosity. 
Phil.  Trans.  (A)  206,  101  (1906);  59  pp. 

BOUSFIELD,  W.  &  LOWRY,  T.  192,  (1)  The  Influence  of  Temperature  on  the 
Conductivity  of  Electrolytic  Solutions.  Proc.  Roy.  Soc.  London  71, 
42  (1902);  13  pp.;  (2)  The  Electrical  Conductivity  and  other  Properties 
of  Sodium  Hydroxide  in  Aqueous  Solutions  as  Elucidating  the  Mech- 
anism of  Conduction.  Proc.  Roy.  Soc.  London  74,  280  (1904);  4  pp. 
Phil.  Trans.  (A)  204,  253  (1905);  69  pp. 

BOUSSINESQ,  J.  6,  18,  50,  (1)  Essai  sur  la  th6orie  des  eaux  courantes. 
M6m.  presentes  par  divers  savants  a  1'Acad.  des  Scien.  23,  (1877); 
680  pp.;  (2)  Additions  et  e"claircissements  au  memoire  institute":  Essai 
sur  la  theorie  des  eaux  courantes.  Do.  24,  (1877);  64  pp.;  (3),  Lec.ons 
synthetiques  de  Mecanique  ge'ne'rale,  servant  d'introduction  au  Cours  de 
Mecanique  physique  de  la  Faculte  des  Sciences  do  Paris.  Gauthier- 
Villars  (1889);  (4)  Theorie  du  regime  permanent  graduellement  varie 
qui  se  produit  pres  de  1'entree  evasee  d'un  tube  fin,  ou  les  filets  d'un 
liquide  qui  s'y  ecoule  n'ont  pas  encore  acquis  leurs  inegalites  normales 
de  vitesse.  Compt.  rend.  110,  1160  (1890);  6  pp.;  (5;  The"orie  de 
mouvement  permanent  qui  se  produit  pres  de  1'entree  evasee  d'un 
tube  fin:  application  a  la  deuxieme  s6rie  d' experiences  de  Poiseuille. 
-  Compt.  rend.  110,  1238  (1890);  5  pp.;  (6)  Sur  1'explication  physique 
dela  fluidite.  Compt.  rend.  112,  1099  (1891);  3  pp.;  (7)  Sur  la  Maniere 
dont  les  vitesses,  dans  un  tube  cylindrique  de  section  circulaire,  evase 
&  son  entree,  se  distribuent  depuis  cette  entree  jus  q'aux  endroits  on 
se  trouve  etabli  un  regime  uniforme.  Compt.  rend.  113,  9  (1891); 
6  pp. ;  (8)  Calcul  de  la  moindre  longueur  que  doit  avoir  un  tube  circu- 
laire, evase  a  son  entree  pour  qu  'un  regime  sensiblement  uniforme  s'y 
e'tablisse  et  de  la  depense  de  charge  qu'y  entratne  1'etablissement. 
Compt.  rend.  113,  49  (1891);  2  pp.;  (9)  Theorie  analytique  de  la  chaleur 
Vol.  II.  Note  I,  196-265.  Sur  la  resistanse  opposee  aax  petits  mouve- 
ments  d'un  fluide  indefini  par  un  solide  immerge'  dans  ce  fluide. 
Gauthier-Villars,  Paris,  1903;  (10)  Existence  of  Surface  Viscosity  in  the 
Thin  Transition  Layer  Separating  a  Liquid  from  a  Contiguous  Fluid. 
Ann.  chim.  phys.  29,  349  (1913);  8  pp.;  (11)  Application  of  Surface 
Viscosity  Formulas  to  the  Surface  of  a  Spheroidal  drop  Falling  with 
Uniform  Velocity  into  a  Fluid  of  less  Specific  Gravity.  Ann.  chim. 
phys.  29,  357  (1913);  7  pp.;  (12)  Velocity  of  the  Fall  of  a  Spherical 
Drop  into  a  Viscous  Fluid  of  Less  Specific  Gravity.  Ann.  chim.  phys. 
29, 364  (1913) ;  7  pp. ;  (13)  Internal  Friction  and  Turbulent  Flow.  Compt. 
rend.  1517  (1896). 


358  INDEX 

BOUTARIC,  A.     Sur  quelques  consequences  physico-chimiques  des  mesures 

de  viscosite.     Rev.  gen.  Sci.  25,  425  (1914);  8  pp. 
BOUTY  and  BENDER.     192. 
BOVEY,  H.     Some  Experiments  on  the  Resistance  to  Flow  of  Water  in  Pipes. 

Trans.  Roy.  Soc.  of  Canada  Sect.  3  (1898);  14  pp. 
BOYNTON,  W.  P.     Application  of  the  Kinetic  Theory  to  Gases,  Vapors,  Pure 

Liquids    and    the    Theory    of    Solutions.      Macmillan     Co.     (1904); 

288  pp. 
BRAUN,  W.     Uber  die  Natur  der  elastischen  Nachwirkung.     Pogg.  Am. 

159,  337  (1876);  62  pp. 
BRAUN,  W.  &  KURZ,  A.     (1)  Uber  die  Dampfung  der  Torsionsschwingungen 

von  Drahten.     Carl's  Repert.     Exp.-physik.  15,  561  (1879);  16  pp.; 

(2)  Do.,   II,   Carl's  Repert.     Exp.-physik.   17,  233   (1881);  21   pp.; 

(3)  Uber  die  elastische  Nachwirkung  in  Drahten.     III.  Carl's  Repert. 
Exp.-physik.  18,  665  (1882);  8  pp. 

BREDIG,  G.  192,  Beitrage  zur  Stochiometrie  der  lonenbeweglichkeit. 
Z.  physik.  Chem.  13,  190  (1894);  98  pp. 

BREITENBACH,  P.  79,  252,  (1)  tlber  die  innere  Reibung  der  Gase  und  deren 
Anderung  mit  der  Temperatur.  Diss.  Erlangen  (1898);  Wied.  Ann. 
67,  803  (1899);  25  pp;  (2)  Do.,  Ann.  Physik.  5,  166  (1901);  4  pp. 

BRIDGMAN,  P.  W.  (1)  Mercury,  Liquid  and  Solid,  Under  Pressure.  Proc. 
Amer.  Acad.  47,  345  (1912);  94pp.;  (2)  On  the  Effect  of  General  Mech- 
anical Stress  on  the  Temperature  of  Transition  of  Two  Phases,  with 
a  Discussion  of  Plasticity.  Phys.  Rev.  (2)  7,  215  (1916);  9  pp. 

BRIGGS,    BENNETT  and    PIERSON.     J.    Phys.    Chem.    22,    256    (1918). 

BRILLOUIN,  M.  32,  38,  42,  142,  (1)  Theorie  elastique  de  la  plasticite  et 
la  fragilite  des  corps  solides.  Compt.  rend.  112,  1054  (1891);  2  pp.; 
(2)  Viscosity  of  Liquids  as  a  Function  of  the  Temperature.  Ann.  chim. 
phys.  (7)  18, 197  (1899) ;  16  pp. ;  (3)  Sur  la  viscosite  des  fluides.  Compt. 
rend.  144,  1151  (1907);  2  pp.;  (4)  Lecons  sur  la  viscosite  des  liquides 
des  gaz.;  Part  I.  Gcneralite"  des  viscosite's  des  liquides.  VII  +  228 
pp.  Part  II.  Viscosite  des  gaz.  Caracteres  generaux  des  th6ories 
mol<§cularies.  146  pp.  Gauthier-Villers.  Paris  (1907);  (5)  Diffusion 
of  Animated  Particles  in  Brownian  Movement.  Ann.  chim.  phys. 
27,  412  (1913);  11  pp.;  (6)  Thermal  Conductivity  and  Viscosity  of 
Monatomic  Liquids.  Compt.  rend.  159,  27  (1914);  3  pp. 

BRINKMAN,  C.  Die  innere  Reibung  als  Hiilfsmittel  zur  Erkennung  und 
Unterscheidung  ahnlich  konstituierter  Verbindungen.  Diss.  Leipzig 
(1903);  54  pp. 

BRITTEN,  R.  P.  L.  Hardness  and  Viscosity  of  Varnishes.  Oil  and  Color 
Chem.  Assoc.  Sept.,  17  (1918);  J.  Soc.  Chem.  Ind.  37,594  (1918). 

BRODMAN,  C.  30,  (1)  Untersuchungen  liber  den  Reibungscoefficienten  von 
Flussigkeiten.  Diss.  Gottingen  (1891);  Wied.  Ann.  45,  159  (1892); 
26  pp. ;  (2)  Uber  eine  zur  Untersuchung  sehr  zaher  Fliissigkeiten  geeig- 
nete  Modification  der  Transpirations  methode.  Wied.  Ann.  46,  188 
(1893);  19  pp.  (Glycerol). 

BROWN,  D.  F.     162,  et  seq.,  168,  Thesis  Lafayette  College  (1921). 


INDEX  359 

BROWN,  G.     (1)  The  Viscosity  of  Some  Shales  at  Furnace  Temperatures. 

Trans.  Am.  Ceram.  Soc.  16,  571  (1914);  5  pp. 
BRUCE,  H.  D.     296,  322,  Thesis  Lafayette  College  (1922). 
BRUCKNER,  H.     6,  63,  179,  Uber  Reibung  von  Salzlosungen.     Diss.  Halle 

(1890);  Wied.  Ann.  42,  287  (1891);  23  pp. 
BRUHL,    J.    W.     Die    Beziehungen    zwischen    der   physikalischen    Eigen- 

schaften  organischer  Korper  und  ihrer  chemischen  Constitution  III. 

Ber.  d.  deutsch  chem.  gesell  13,  1529  (1880);  2  pp. 
BRUNHES,  J.  &  DUSSY,  J.     Sur  les  variations  de  viscosite  que  presente  le 

soufre  fondu.     Compt.  rend.  118,  1045  (1894);  2  pp. 
BUBANOVIC  F.     The  Influence  of  Fat-Soluble  Materials  on  the  Viscosity 

and  Surface  Tension  of  Olive  Oil.     Z.  Chem.  Ind.  Kolloide  10,  178 

(1912);  3  pp. 
Bucco,    M.     Viscometric  Investigations  in   Relation  to   Blood   Pressure. 

Atti  XXIII  Congr.  med.  inst.  Roma  373,  Zentr.  Biochem-Bio-phys. 

16,  196. 
BUCHANAN,  J.  and  MALCOLM,  H.  W.     Experiments  with  Rotating  Viscous 

Liquids.     Phil.  Mag.  (6)  9,  251  (1905);  7  pp. 
BUCHBOCK,  G.     (1)  Uber  die  Geschwindigkeit  hydrolytischen  Zerzetzung 

der  Karbonylsulfids.     Z.  p.  ch.  23,  123  (1897)  34  pp.;  (2)  Uber  den 

Einfluss  des  Mediums  auf  die  Reactions.     Z.  physik.  chem.  34,  229 

(1900);  19pp. 
BUCKINGHAM,  E.     223,  et  seq.,  (1)  Studies  on  the  Movement  of  Soil  Moisture. 

U.  S.  Bureau  of  Soils  Bull.  #3841;  (2)  Model  Experiments  and  the  Forms 

of  Empirical  Equations.     Trans.   Am.  Soc.   Mech.  Engineers  (1915); 

(3)  On  Plastic  Flow  through  Capillary  Tubes.     Proc.  Am.  Soc.  Testing 

Materials  (1921). 
BUHNER,  C.     209,  Beitrage  zur  Kentniss  der  Kristallinischen  Fliissigkeiten. 

Inaug.  dissert.,  Marburg  (1906);  46  pp. 
BUGLIA,   C.     211,    (1)    Uber  einige  physikalisch-chemisch   Merkmale  der 

homogenisierten  Milch.     Kolloid-Z.  2,  353  (1908). 
BURKHARD.     Diss.  Zurich,  Berlin  (1873);  Z.  Rubenzuckerindustrie  (1874); 

99. 

BURRI,  R.  and  NUSSBAUMER,  T.     Surface  Tension  and  Viscosity  Determi- 
nations on  Milk  by  Means  of  the  Traube  Stalagmometer.     Biochem. 

Z.  22,90(19— );  13  pp. 
BURTON-OPITZ,  R.     286,  (1)  Uber  die  Veranderung  der  Viskositat  des  Blutes 

unter    dem    Einfluss    verschiedener    Ernahrung  und  experimentaller 
Eingriffe.     Arch,   ges  Physiol.    (Pfltiger's)  82,  447  (1900);  17  pp.;  (2) 

Vergleicb   der  Viskositat  des  normalen   Blutes  mit  der  des  Oxalat- 

blutes,    des    defibrinirtes    Blutes   und    Blutserums   bei   verschiedener 

Temperatur.     Pfluger  Arch.  82,  464  (1900);  10  pp.;  (3)  The  Viscosity 

ofLaked  Blood.     Am.  J.  Physiol.  35,  (1914);  7pp.;  (4)  The  Viscosity 

of  Bile.     Biochem.  Bull.  3,  351  (1914);  1  p.;  (5)  J.  Physiol.  32,  (1904); 

(6)  Arch.  ges.  Physiol.  (Pfliiger's)  119  (1907). 
BURTON-OPITZ,  R.  and  DINEGAR,  R.     The  Viscosity  of  Urine.     Am.  J. 

Physiol.  47,  220  (1918);  11  pp. 


360  INDEX 

BUTCHER,  J.     216,  On  Viscous  Fluids  in  Motion.     Proc.  London  Math. 
Soc.  8,  103  (1876);  33  pp. 

CALLAN,  J.  G.     A  Viscosity  Indicator  for  Paper-Making  "Beater-Stock." 

U.  S.  Pat.  1,331,861. 
CAMERER,  R.     O'lreibung  in  Rohren.     Zeitschr.    gesamte    Turbinenwesen 

4,  461  (1907);  6  pp. 
CAMPANI,  A.  &  LEOPARDI.     Viscosity  of  the  Blood  and  the  Use  of  Alkalies. 

Folia  clin.  4,  91  (1912);  9  pp. 
CAMPBELL,  H.     The  Resistance  to  the  Blood  flow.     J.  Physiol.  23,  301 

(1898-1899). 
CANTONE,    M.     Elastic    Hysteresis.     Rend.    Ace.    Lincei    4,    437    (1895); 

9pp. 
CAPPENBERG,  H.     A  Viscometer  for  Comparison  of  Hot  Pastes.     Chem. 

Ztg.  34,  218  (1909). 
CARLSON,  O.  and  THALL,  E.     Cellulose  Ester  Solutions.     Brit.  Pat,  136,141 

Sept.  9  (1919). 
CAROTHERS,  S.  D.     Portland  Experiments  on  the  Flow  of  Oil  in  Tubes. 

Proc.  Roy.  Soc.  (London)  (A)  87,  154  (1912);  9  pp. 
CARPENTER,  C.  E.     Viscosity  in  Cylinder  Oils.     Power  43,  519   (1916); 

Also  Do.,  44,  101,249,  463,631,  697,598,  791  (1916);  10  pp. 
CARPINI,  C.     35,  Veranderung  der  inneren  Reibung  magnetischen  Fliissig- 

keiten  im  magnetischen  Felde.     Rend.  R.  ace.  des  Line.  12,  2  Sem. 

341  (1903);  13  pp.     Beible  27,  1127  (1903). 
CAVAZZANI,  E.     286,  (1)  Viscosita  degli  umori  dell'occhio.     Arch,  di  farmac. 

speriment.  e  scienze  affini  4,  401  (1905);  (2)  Reazione  viscosimetrica 

del  latte.     Archiv.  di  Fisiol.  2,  513  (1905);  (3)  Viscositat  der  Milch. 

Zentralblatt  f.  Physiol.  18,  841  (19—). 
CECONI,  A.     II  metodo  della  conducibilita  elettrica  nell'indagine  clinica. 

La  clinica  medica  italiana  44,  635  (1905). 
CERVELLO  and  PITINI.     Sulle  variazioni  termiche  della  viscosita  dei  colloidi. 

Arch,  di  farmacologia  e  terapeutica  12,  17  (1906). 
CHANDELON,  T.     Viscosity  of  Collodion.     Bull.   soc.   chim.  belg.   28,  24 

(1914);  8  pp.;  J.  Soc.  Chem.  Ind.  33,  222  (1913). 
CHANOZ,  M.     Dissymmetry  Produced  by  a  Direct  Current  at  Symmetrically 

Arranged  Liquid  Junctions;  Effect  of  Viscosity.     Compt.  rend.   149, 

598  (1910);  2  pp. 
CHAPMAN,  S.     (1)  Kinetic  Theory  of  Simple  and  Composite  Monatomic 

Gases:   Viscosity,   Thermal  Conduction,   and   Diffusion.     Proc.   Roy. 

Soc.  London  (A)  93, 1  (1916);  20  pp.;  (2)  Law  of  Distribution  of  Molecu- 
lar Velocities  and  the  Theory  of  Viscosity  and  Thermal  Conduction 

in  a  Non-Uniform  Simple  Monatomic  Gas.  Trans.  Roy.  Soc.  London 

216,279  (1916);  68  pp. 
CHARBONNIER,  P.     La  the"orie  du  champ  acoustique  et  le  frottement  inte'- 

rieur  des  Gaz.  Compt.  rend.  137,  378  (1903);  3  pp. 
CHEBOKSAROV,  M.     Influence  of  Iodine  Compounds  on  the  Viscosity  of  the 


INDEX  361 

Blood.     Z.   exp.   Med.   2,   168   (1913);  6  pp.;     Cp.  Zentr.  Biochem. 

Biophys.  16,  57. 
CHELLA,  S.     (1)   Apparat  zur  absoluten  Messung  der  Koefficienten  der 

inneren  Reibung  der  Gase.     Physik.  Z.  7,  196  (1906);  3  pp.;  (2)  Mes- 
sung   des   inneren    Reibung    der   Luft   bei    niedrigen    Temperaturen. 

Physik.  Z.  7,  546  (1906);  2  pp. 
CHENEVEATJ,  G.     (1)  Solutions  of  Et  OH,  H2SO4,  HNO3;  Hydrates  Et  OH.- 

3H2O,  H2SO<.   H2O,  HNO3.  2  H2O.     Compt.  rend.  156,  154  (1912); 

1  p.;   (2)   The  Measurement  of  the  Viscosity  of  Oils.     J.  physique  7, 

109  (1917);  6  pp. 
CHEVENARD,  P.     The  Viscosity  of  Steels  at  Elevated  Temperatures.  Compt. 

rend.  169,  712  (1919);  5  pp. 
CHICK,    H.    &   LUBRZYNSKA.     (1)    Viscosity    of    Some    Protein  Solutions. 

Biochem.  J.  8,  59  (1913);  10  pp.;  (2)  Pseudoglobulin  and  Euglobulin 

(Horse).     Do.  8,  261  (1914);  19  pp. 
CHICK,  H.  &  MARTIN,  C.  J.     The  Viscosity  of  Casein  Sols.     Z.  Chem.  Ind. 

Kolloide  11,  102  (1912);  3  pp. 
CHOROWER,   C.     Behavior  of  Varieties  of  Casein   (from  Cow  and  Goat 

Milk)  with  Reference  to  Causing  Viscosity.     Chem.  Ztg.  44,  605,  613 

(1920);  4  pp. 
CHOWNE,  W.     Experimental  Researches  on  the  Movement  of  Atmospheric 

Air  in  Tubes.     Proc.  Roy.  Soc.  London  7,  466  (1855);  13  pp. 
CHRISTIANSEN,  C.     Versuche  liber  den  Einfluss  der  Capillaritat  auf  die 

Ausstromungsgeschwindigkeit    des    Flussigkeiten.     Ann.    Physik    (4) 

5,  436  (1901);  11  pp.     Cp.  Wied.  Am.  41,  565  (1890). 
CHWOLSON.     Reibung  im  Innern  der  Flussigkeiten.     Lehrbuch  der  Physik., 

Vol.  I,  Braunschweig  (1902). 
CLAMER,  G.  H.     A  Study  of  Alloys  Suitable  for  Bearing  Purposes.     Frank. 

Inst.  166,  49  (1903);  29  pp. 
CLARK,  R.     Variation  of  the  Viscosity  of  Gases  with  the  Temperature. 

Trans.  Roy.  Soc.  Canada  13,  III,  177  (1919);  4  pp. 
CLEBSCH,  A.     Theorie  der  Elasticitat  fester  Korper.     Leipzig  (1862). 
CLEMENT,  L.     Theory  of  Viscometers.     Matieres  grasses  2,  1591  (1909);  2 

pp. 
CLERICI,  E.     Viscosity  of  Liquids  for  the  Mechanical  Separation  of  Minerals. 

Atti.  accad.  Lincei  20,  I,  45  (1910);  6  pp. 
CLULOW,   F.   S.   and  TAYLOR,   C.   W.     Consistency  of  Greases.     J.   Soc. 

Chem.  Ind.  39,  291  (1920). 
CMUNT,  E.     Action  of  the  Ingestion  of  Gelatin  on  the  Viscosity  of  the  Blood. 

Med.  Klinik  8,  1393  (1913);  2  pp. 
COHEN,  R.     138,  139,  140,  tlber  den  Einfluss  des  Druckes  auf  die  Viskosi- 

tat  von  Flussigkeiten.     Wied.  Ann.  45,  666  (1892);  19  pp. 
COKER,    E.     The    critical  velocity   of   flow    of   mercury   in    small  tubes. 

Engineering  94,  581  (1914). 
COKER,  E.  &  CLEMENT,  S.     48,  An  Experimental  Determination  of  the 

Variation  of  the  Critical  Velocity  of  Water  with  the  Temperature. 


362  INDEX 

Phil.  Trans.  A  201,  45  (1903);  16  pp.;  Proc.  Roy.  Soc.  London  71, 

152  (1903);  1  p. 
COLEMAN,  J.     Notes  on  Viscosity  and  other  Tests  for  Oils.     J.  Soc.  Chem. 

Ind.  5,359  (1886);  2  pp. 
COLIN,  H.  and  CHAUDUN,  A.     Law  of  Action  of  Sucrose.     Influence  of 

Viscosity  upon  the  Speed  of  Hydrolysis.     Compt.   rend.    168,   1274 

(1919);  2  pp. 
COLLINS,  A.  L.     Testing  the  Viscosity  of  Liquids.     U.  S.  Pat.  1,224,142, 

May  1  (1917). 
COLSON,  A.     Sur  1'ecoulement  des  liquides  en  tubes  capillaires.     Compt. 

rend.  113,740  (1891);  3  pp. 
COMBO,  E.     Sulla  resistenza  dei  corpuscoli  rossi  di  fronte  a  soluzioni  colloid- 

ali.     Lo  Sperimentale  67,  331  (1903). 
COUETTE,  M.     6, 17, 22,  et  seq.  29,  33,  42,  et  seq. ;  (1)  La  viscosite  des  liquides. 

Bulletin  des  Sciences  physiques  (1888);  (2)  Sur  un  nouvel  appareil 

pour  1'etude  du  frottement  des  fluides.     Compt.  rend.  107,  388  (1888); 

3  pp.;  (3)  Etudes  sur  la  frottement  des  liquides.     Ann.  chim.  phys. 

(6)  21,  433  (1890);  72  pp.;  (4)  Distinction  de  deux  r6gimes  dans  le 

mouvement  des  fluides.      J.  de  physique  (2)  9,  414;  560  (1890);  11  pp. 
COULOMB.     1,  6,  29,  138,  261,  et  seq.;  (1)  Hist,  de  I'Academie  251  (1784); 

(2)  Exp6riences  destinies  a  determiner  la  coherence  des  fluides  et  les 

lois  de  leur  resistance  dans  les  mouvemens  tres-lents.     Mem.  de  1'Inst. 

nat.  des.  Scienc.  et  Arts.  Scienc.  Math,  et  Phys.  3,  246  (1801);  60  pp. 
COUPLET.     1,  Recherches  sur  le  mouvement  des  eaux.     Histoire  de  1'Acad. 

royale  des  Sciences  113  (1732);  55  pp. 
COWLEY,    W.    L.     Fluid    Motion    and    Viscosity.     Engineering    109,    101 

(1920);  2  pp. 
CROOKES,  W.     245,  (1)  On  the  Viscosity  of  Gases  at  High   Exhaustions. 

Phil.  Trans.  B.  172,  387  (1881);  48  pp.;  Proc.  Roy.  Soc.  London  31, 

446  (1881);  13pp. 
CUNNINGHAM,  E.     On  the  Velocity  of  Steady  Fall  of  Spherical  Particles 

through  a  Fluid  Medium.     Proc.  Roy.  Soc.  83A,  357  (1910);  9  pp. 
CUSHMAN,    A.    S.     (1)     The  Colloid     Theory    of  Plasticity.     Trans.  Am. 

Ceram.  Soc.  6,  65  (1904),  14  pp.;  (2)  On  the  cause  of  the  cementing 

value  of  rock  powders  and  the  plasticity  of  clays.     J.  Am.  Chem.  Soc. 

26,  451  (1903);  18  pp.;  (3)  Extraction  of  Potash  from  feldspathic  rock. 

J.  Am.  Chem.  Soc.  30,  779  (1908);  (4)  Bull.  U.  S.  Dept.  Agric.  92 

(1905). 
CZERNY,  A.     Versuche  iiber  Bluteindickung  und  ihre  Folgen.     Arch.  exp. 

Path.  Pharm.  34,  268  (1894). 

D'ALEMBERT.     1,    Traite"    de   l'6quilibre   et   du    mouvement   des   fluides. 

Nouvelle  e"dit.,  Paris  (1770). 
DARCY,  H.     1,  36,  Recherches  expe>imentales  relatives  au  mouvement  dc 

1'eau  dans  les  tuyaux.     Paris   (1857);   M6m.  par  divers,  savants  a 

1'Acad.  des  Scienc.  de  1'Inst.  15,  141  (1858);  263  pp.     Cp.  Memoiren 

der  kaiser.  Akademie  der  Wissenschaften  16. 


INDEX  363 

D'ARCY,  R.     Viscosity  of  Solutions,  Phil.  Mag.  (5)  28,  221  (1889);  11  pp. 

DAVIDSON,  G.  F.  Flow  of  Viscous  Fluids  through  Orifices.  Proc.  Roy. 
Soc.  London  (A)  89,  91  (1913);  8  pp. 

DAVIS,  N.  B.  The  Plasticity  of  Clay.  Trans.  Am.  Ceram.  Soc.  16,  65 
(1914);  15pp. 

DAVIS,  P.  B.,  ET  AL.  (1)  Studies  on  Solution  in  its  Relation  to  Light 
Absorption,  Conductivity,  Viscosity  and  Hydrolysis.  Carnegie  Inst. 
of  Washington,  D.  C.,  144  pp.;  (2)  A  Note  on  the  Viscosity  of  Caesium 
Salts  in  Glycerol-Water  Mixtures.  Carnegie  Inst.  Pub.  260,  97 
(1918);  1  p.;  (3)  The  Conductivity  and  Viscosity  of  Organic  and 
Inorganic  Salts  in  Formamide  and  in  Mixtures  of  Formamide  with 
Ethyl  Alcohol.  Carnegie  Pub.  260,  71  (1918);  26  pp. 

DAVIS,  P.  B.,  HUGHES,  H.  &  JONES,  H.  C.  Conductivity  &  Viscosity  of 
Rubidium  Salts  in  Mixtures  of  Acetone  &  Water.  Z.  physik.  Chem. 
85,513  (1913);  39  pp. 

DAVIS,  P.  B.  and  JONES,  H.  C.  Conductivity  and  Negative  Viscosity 
Coefficients  of  Certain  Rubidium  &  Ammonium  Salts  in  Glycerol  and 
in  Mixtures  of  Glycerol  and  Water  at  25°  to  75°.  Z.  physik.  Chem. 
81,  68  (1913);  45  pp. 

DAWSON,  H.  M.  The  Estimation  of  Mixtures  of  Isomers  and  Other 
Closely  Related  Substances.  J.  Soc.  Dyers  Colourists  35,  123  (1919); 
6pp. 

DAY,  H.  The  Effect  of  Viscosity  on  Thermal  Expansion.  Am.  J.  Sci. 
(4)  2,  342  (1896);  5  pp.;  Nature  55,  92  (1896-7). 

DEAN,  E.  W.  and  JACKSON,  L.  E.  Effect  of  crystalline  paraffine  wax  upon 
the  viscosity  of  lubricating  oil.  U.  S.  Bur.  Mines  Reports  of  Investi- 
gations, #2249  (1921);  3  pp. 

DEELEY,  R.  M.  Oiliness  and  Lubrication.  Engineering  108,  788  (1919); 
Proc.  Phys.  Soc.  London,  II  28,  11  (1919);  Do.,  II  32,  1  (1920);  11  pp. 
(Discussion  by  Deeley,  Martin,  Allen,  Skinner,  Southcombe,  and 
Hardy.) 

DEELEY,  R.  M.  &  PARR,  P.  H.  239,  (1)  The  Viscosity  of  Glacier  Ice. 
Phil.  Mag.  (6)  26,  85  (1913);  26  pp.;  (2)  The  Hintereis  Glacier.  Phil. 
Mag.  (6)  27,  153  (1914);  24  pp. 

DEERING,  W.  H.  &  REDWOOD,  B.  Report  on  Castor  Oils  from  Indian 
Section  of  the  Imperial  Inst.  J.  Soc.  Chem.  Ind.  13,  959  (1894);  2  pp. 

DE  GUZMAN,  J.     Anales  Soc.  Expan.  fis.  quim  11,  353  (1913);  9  pp. 

DENISON,  R.  B.  172,  Liquid  Mixtures,  II.  Chemical  Combination  in 
Liquid  Binary  Mixtures  as  Determined  in  a  Study  of  Property  Com- 
position Curves.  Trans.  Faraday  Soc.  8,  35  (1913). 

DENNHARDT,  R.     Ann.  Phys.  67,  325  (1899). 

DEWAR,  J.  The  Viscosity  of  Solids.  Nature  50,  238  (1894);  Chem.  News 
69,  307  (1894);  1  p. 

DETERMANN,  H.  (1)  Ein  einfaches,  stets  gebrauchfertiges  Instrument  zur 
Messung  der  Inneren  Reibung  von  Fliissigkeiten.  Physik.  Z.  9, 
375  (1908);  1  p.;  Munich,  Med.  Wochenschr.  #42  (1907);  (2)  Viscosity 
and  Protein  Content  of  the  Blood  with  Different  Diets  Especially  with 


364  INDEX 

Vegetarians.     Med.  Klin.   No.  24   (1909);  Berlin,  klin.   Wochenschr. 

664  (1909).     Cp.  Munich  Med.  Wochenschr.  #23  (1907). 
DETERMANN,  H.  &  BOOKING.     Does  the  Administration  of  Iodine  Influence 

the  Viscosity  of  the  Blood?    Deut.  Med.  Wochschr.  38,  994  (1912);  lp. 
DETERMANN,  H.  and  WEIL,  F.     Viscosity  and  Gas  Content  of  Human 

Blood.     Z.  Klin.  Med.  70,  468  (1911);  6  pp. 
DICKENSCHEID,    F.     209,    Untersuchungen    iiber    Dichte,    Reibung,    und 

Kapillaritat  kristallinischer  Flussigkeiten.     Diss.  Halle  (1908);  44  pp. 
DIENES,  L.     Viscosity  of  Colloidal  &  Non-Colloidal  Liquids.     Biochem. 

Z.  33,  222  (1910);  3pp. 
DOELTER,  C.     287,   (1)  Silikatglaser  und  Silikatschmelzen.     Chem.  Tech. 

Ztg.   (2)  9,  76  (1906);  Sitzber.  Wien.  Akad.  114,  I,  529  (1905);   (2) 

tJber     den     Einfluss     der     Viskositat    bei    Silikatschmelzen.     Centr. 

Min.  193   (1906);   (3)  The  Viscosity  of  Silicate  Melts.     Chem.  Ztg. 

36,  569  (1913). 
DOELTER,  C.  &  SIRK,  H.     The  Determination  of  the  Absolute  Value  of  the 

Viscosity  of  Silicate  Melts.     Monatsber.  32,  643  (1912);  Sitzber.  Wien. 

Akad.  20,  I,  659  (1911). 
DOLAZALEK,  F.  und  SCHULZE,  A.     Zur  Theorie  der  binaren  Gemische  und 

konzentrierten    Losungen    IV   Das  Gemisch:  Athylather-Chlo  reform. 

Z.  physik.  Chem.  83,  45  (1913);  34  pp. 
DOLFUS,  C.     Bull.  Soc.  Ind.  Mulhouse  5,  14-23. 

DONLINSON,  H.     The  Influence  of  Stress  and  Strain  on  the  Physical  Prop- 
erties of  Matter.     Phil.  Trans.  London  177,  Pt.  II,  801  (1886);  37  pp. 
DONNAN,  F.     The  Relative  Rates  of  Effusion  of  Argon,  Helium  and  Some 

other  Gases.     Phil.  Mag.  (5)  49,  423  (1900);  23  pp. 
DOOLITTLE,  O.     328,  The  Torsion  Viscometer.     J.  Am.  Chem.  Soc.  16, 

173  (1893);  5  pp.;  J.  Soc.  Chem.  Ind.  12,  709  (1893);  1  p. 
DORN,  E.  &  VOLLMER.     Uber  die  Einwirkung  von  Salzsaure  auf  metallisches 

Natrium  bei  niederen  Temperaturen.     Ann.  Physik.  60,  468  (1897);  10 

pp. 
DOROSHEVSCHII,  A.  G.  and  ROZHDESTORNSKII.     195,  Electrical  Conductivity 

of  Mixtures  of  Alcohol  and  Water.     J.  Russ.  Phys.  Chem.  Soc.  40, 

887  (1908);  21  pp. 
DOWLING,  J.  J.     Steady  and  Turbulent  Motion  in  Gases.     Proc.    Roy. 

Soc.  Dublin  13,  375-98  (1913);  23  pp. 
DRAPIER,  P.     98,  99,  100,  102,  103,  Viscosity  of  Binary  Liquid  Mixtures  in 

the  Neighborhood  of  the  Critical  Solution  Temperature.     Bull.  acad. 

roy.  belg.  1,  621  (1911);  19  pp. 
DREW,  E.     A  Determination  of  the  Viscosity  of  Water.     Physik.  Rev. 

12,  114  (1901);  7  pp. 
DRXJCKER,    K.     Fluidity.    I.   Z.  physik.   Chem.  92,  287   (1917);  32  pp.; 

J.  Chem.  Soc.,  II  112,  409. 
DRUCKER,   K.   &  KASSEL.     104,  The  Fluidity  of  Binary   Mixtures.     Z. 

physik.  Chem.  76,  367  (1911);  18  pp. 
DUBRISAY,  R.     A  Method  of  Testing  the  Viscosity  of  Lubricating  Oils. 

Ann.  fals,  10,  301  (1917);  4  pp.;  J.  Soc.  Chem.  Ind.  36,  1123  (1917). 


INDEX  365 

DuBuAT,  C.  1,  Principes  d'hydraulique  verifies  par  un  grand  nombre 
d'experiences.  nouv.  <§dit.,  Paris  (1786). 

DUCCESCHI,  V.  (1)  I  processi  di  ossidazione,  di  riduzione  e  di  sintesi  negli 
animali  stiroidati.  Sperimentale  50  (1897);  (2)  Sui  proteici  del  siero 
sanguigno  nei  cani  animizzati  e  stiroidati.  Morgagni  Sez.  biologica  1, 
(1897). 

DTJCLAUX,  E.  Recherches  zur  les  lois  des  mouvements  des  liquides  dans  les 
espaces  capillaires.  Ann.  chim.  phys.  (4)  23,  433  (1872);  69  pp. 

DUCLAUX,  J.  &  Miss  WOLLMAN.  281,  Recherches  sur  la  cellulose  et  ses 
ethers.  Bull.  Soc.  Chim.  (4)  27,  414  (1920);  6  pp. 

DUDETZKII.  235,  Viscosity  of  Asphalt.  J.  Russ.  Phys.  Chem.  Soc., 
Phys.  Pt.  45,  519  (1914);  13  pp. 

DUFF,  W.  6,  34,  131,  (1)  The  Viscosity  of  Polarized  Dielectrics.  Physik. 
Rev.  4,  23  (1896);  16  pp.;  (2)  Empirical  Formulae  for  Viscosity  as  a 
Function  of  the  Temperature.  Physik.  Rev.  4,  404  (1896);  6  pp.; 
(3)  Viscosity  of  Liquids  at  Low  Rates  of  Shear.  Science  (N.  S.)  17, 
184  (1903);  1  par.;  (4)  Poiseuille's  Law  at  Very  Low  Rates  of  Shear. 
Phil.  Mag.  (6)  9,  685  (1905);  7  pp. 

DUHEM,  P.  (1)  Sur  les  fluides  compressibles  visqueux.  Compt.  rend.  134, 
1088  (1902);  3  pp.;  (2)  La  viscosite  au  voisinage  de  1'etat  critique. 
Compt.  rend.  134,  1272  (1902);  3  pp.;  (3)  Sur  la  viscositS  en  un  milieu 
vitreux.  Compt.  rend.  136,  281  (1903);  3  pp.;  (4)  Des  ondes  du  premier 
ordre  par  rapport  a  la  vitesse  au  sein  d'un  milieu  vitreux  dou£  de 
viscosit^,  et  affect^  de  mouvement  finis.  Compt.  rend.  136,  858 
(1903);  3  pp.;  (5)  Des  ondes  du  second  ordre  par  rapport  a  la  vitesse 
au  sein  des  milieux  vitreux,  doue"  de  viscosit^,  et  affecte  de  mouvement 
finis.  Compt.  rend.  136,  1032  (1903);  3  pp.;  (6)  Stabilite"  et  viscosite". 
Mem.  de  Bordeaux  (6)  3,  121  (1903);  19  pp.;  (7)  Recherches  sur 
T&asticite.  Ann.  de  1'ecole  norm.  (3)  21,  99  (1904);  40  pp.;  (8)  Do., 
Ann.  de  l'e"cole  norm.  (3)  21,  375  (1904);  39  pp.;  (9)  Sur  les  proprietes 
bes  systemes  affectes  a  la  fois  d'hysteresis  et  de  viscosite\  Compt. 
rend.  138,  942  (1904);  3pp.;  (10)  Effet  des  petites  oscillations  de  1'action 
exte>ieure  sur  les  systemes  affecte's  d'hysteresis  et  de  viscosit^.  Compt. 
rend.  138,  1075  (1904);  1  p.;  (II)  Effet  des  petites  oscillations  de  la 
temperature  sur  un  systeme  affecte  d'hyste"resis  et  de  viscositS.  Compt. 
rend.  138,  1196  (1904);  3  pp.;  (12)  Sur  la  viscosite  et  le  frottement  au 
contact  de  deux  fluides.  Proces-verb.  de  Bordeaux  (1902-03),  27 
(1903);  3  pp.;  (13)  Les  conditions  aux  limites.  Le  theoreme  de 
Lagrange  et  la  viscosit^.  Les  coefficients  de  viscosit6  et  la  viscosit^  au 
voisinage  de  I'Stat  critique.  Recherches  sur  1'hydrodynamique,  2  series. 
Paris,  Gauthier-Villers  (1904). 

DUMANSKII,  A.,  ZABOTINSKII,  E.  &  EVSEYER,  M.  A  Method  for  Deter- 
mining the  Size  of  Colloidal  Particles.  Z.  Chem.  Ind.  Kolloide  12, 
6  (1913);  5pp. 

DUMARESQ,  F.  The  Viscosity  of  Cream.  Proc.  Roy.  Soc.  Victoria  (2) 
25,  307  (1913);  15  pp.;  Expt.  Sta.  Record  30,  170. 

DUNCAN,  J.  &  GAMGEE,  A.     Notes  of  some  Experiments  on  the  Rate  of 


366  INDEX 

Flow  of  Blood  and  some  other  Liquids  through  Tubes  of  Narrow 
Diameter.  J.  Anat.  and  Physiol.  5,  155  (1871);  8  pp. 

DUNSTAN,  A.  81,  88,  92,  178,  (1)  Viscosity  of  Liquid  Mixtures.  J.  Chem. 
Soc.  86,  817  (1904);  10  pp.;  Z.  physik.  Chem.  49,  590  (1904);  7  pp.; 
(2)  Do.,  Part  II.  J.  Chem.  Soc.  87, 11  (1905);  6  pp.;  Z.  physik.  Chem. 
61,  732  (1905);  7  pp.;  (3)  Do.  Part  III.  Proc.  Chem.  Soc.  22,  89 
(1906);  1  p.;  (4)  Innere  Reibung  von  Fliissigkeitgemischen.  Z. 
physik.  Chem.  66,  370  (1906);  10  pp.;  (5)  The  Viscosity  of  Sulphuric 
Acid.  Proc.  Chem.  Soc.  30,  104  (1914);  2  pp.  Cp.  Thole,  Hilditch 
and  Mussel;  (6)  Viscosity  of  Solutions  and  its  Bearing  on  the  Nature 
of  Solution.  J.  Soc.  Chem.  Ind.  28,  751  (1910).  Cp.  Hilditch,  and 
Thole. 

DUNSTAN,  A.  &  JEMMETT,  W.  Preliminary  Note  on  the  Viscosity  of  Liquid 
Mixtures — Ethyl  Acetate  &  Benzene,  Benzene  &  Alcohol,  Alcohol  & 
Water.  Proc.  Chem.  Soc.  19,  215  (1903);  1  p. 

DUNSTAN,  A.  E.  &  HILDITCH,  T.  P.  Relations  between  Viscosity  and 
other  Physical  Properties.  II.  Influence  of  Contiguous  Unsaturated 
Groups.  J.  Chem.  Soc.  102,  II,  435.  Z.  Elektrochemie  18,  185 
(1913);  4  pp. 

DUNSTAN,  A.  E.  &  LANGTON,  H.  112,  Viscometric  Determination  of 
Transition  Points.  J.  Chem.  Soc.  101,  418  (1912);  7  pp.;  Proc.  Chem. 
Soc.  28,  14  (1912). 

DUNSTAN,  A.  E.  &  MUSSEL,  A.  G.  (1)  The  Application  of  Viscometry  to 
the  Measurement  of  the  Rate  of  Reaction.  J.  Chem.  Soc.  99,  565 
(1911);  7  pp.;  Proc.  Chem.  Soc.  27,  59  (1911);  (2)  Viscosity  of  Certain 
Amines.  J.  Chem.  Soc.  97,  1935  (1910);  10  pp.;  Proc.  Chem.  Soc. 
26,  201  (1910). 

DUNSTAN,  A.  E.  &  STEVENS,  J.  E.  The  Viscosity  of  Lubricating  Oils. 
J.  Soc.  Chem.  Ind.  31,  1063  (1913);  1  p. 

DUNSTAN,  A.  &  STUBBS,  J.  The  Relation  between  Viscosity  and  Chem. 
Constitution.  Pt.  III.  J.  Trans.  Chem.  Soc.  93,  1919  (1909);  8  pp. 

DUNSTAN,  A.  &  THOLE,  F.  B.  Ill,  121, 142, 279,  (1)  The  Relation  Between 
Viscosity  and  Chemical  Constitution.  Part  IV.  Viscosity  and  Hydra- 
tion  in  Solution.  J.  chim.  phys.  7,  204  (1909);  J.  Chem.  Soc.  96,  1556 
(1909);  6  pp.;  Proc.  Chem.  Soc.  25,  219  (1909);  (2)  Relation  between 
Viscosity  and  Chemical  Constitution.  V.  Viscosity  of  Homologous 
Series.  Proc.  Chem.  Soc.  28,  269  (1913);  VI.  Viscosity  an  Additive 
Function.  J.  Chem.  Soc.  103,  129  (1913);  4  pp.;  Proc.  Chem.  Soc. 
28,  269  (1913);  VII.  Effect  of  Relative  Position  of  Two  Unsaturated 
Groups  on  Viscosity.  With  P.  Hilditch.  J.  Chem.  Soc.  103,  133 
(1913);  11  pp.;  Proc.  Soc.  28,  269  (1913);  (3)  The  Existence  of  Racemic 
Compounds  in  Solution.  Trans.  Chem.  Soc.  97,  1249  (1910);  8  pp.; 
(4)  The  Viscosity  of  Liquids  (1914);  VIII  +  92  pp.,  Longmans  Green 
&  Co.  Monographs  of  Inorganic  &  Physical  Chem.  edited  by  Alexander 
Findlay;  (5)  The  Relation  between  Viscosity  and  the  Chemical  Con- 
stitution of  Lubricating  Oils.  J.  Inst.  Petroleum  Tech.  4,  191  (1918); 
38  pp.;  Petroleum  Review  38,  245,  267  (1918);  3  pp. 


INDEX  367 

DUNSTAN,  A.,  &  THOLE,  F.  B.  &  HUNT.  Relations  between  Viscosity  and 
Chemical  Constitution.  Proc.  Chem.  Soc.  23,  207  (1907);  J.  Chem. 
Soc.  91,  1728  (1907);  8  pp. 

DUNSTAN,  A.,  THOLE,  P.  and  BENSON,  P.  The  Relation  between  Viscosity 
and  Chemical  Constitution.  Part  VIII.  Some  Homologous  Series. 
Trans.  Chem.  Soc.  105,  782  (1914);  12  pp.;  Proc.  Chem.  Soc.  29,  378. 

DUNSTAN,  A.  &  WILSON,  R.  (1)  The  Viscosity  of  Liquid  Mixtures.  Proc. 
Chem.  Soc.  22,  308  (1907);  J.  Chem.  Soc.  91,  83  (1907);  9  pp.;  (2)  The 
Viscosity  of  Fuming  Sulphuric  Acid.  J.  Chem.  Soc.  93, 2179  (1908) ;  2  pp. 

DUPERTHIUS,  H.  Contribution  a  1'etude  des  dissociants  autres  que  1'eau. 
Conductibilites  limites.  Viscosite".  Chaleurs  de  dissociant.  Diss.  Laus- 
anne (1908);  47  pp. 

Du  PRE  DENNING.     Cp.  Pre  Denning. 

DURHAM,  T.  C.     54,  201  et  seq.  208.     Cp.  Bingham  and  Durham. 

DUSHMAN,  S.  Methods  for  the  Production  and  Measurement  of  High 
Vacua.  Part  XL  Temperature  Drop,  Slip,  and  Concentration  Drop 
in  Gases  at  Low  Pressures.  Gen.  Electric  Rev.  24,  890  (1921);  11  pp. 

DUTOIT,  P.  &  DUPERTHIUS.  195,  Relations  qui  existent  entre  les  conduc- 
tibilite's  limites  et  la  viscosite.  Arch.  sci.  phys.  nat.  25,  508  (1908); 
1  p.;  J.  chim.  phys.  6,  726  (1909);  5  pp. 

DUTOIT,  P.  &  FREDERICK,  L.  Sur  la  conductibilit6  des  electrolytes  dans 
les  dissolvants  organiques.  Bull.  soc.  chim.  (3)  19,  321  (1898);  17  pp. 

DYSART,  A.  S.     A  Viscometer.     U.  S.  Pat.  1,292,276. 

EDELEANN  L.  &  DULUGEA,  S.     Determination  of  Viscosity  of  Oil  in  Engler 

Viscosimeter.     Petroleum  6,  196  (191-):  1  p. 
EGER,    H.     Untersuchungen    liber   das    Durchstromen   von    Gases   durch 

Kapillaren  bei  niederen  Drucken.     Ann.  Physik.  (4)  27,  819  (1908); 

24pp. 
EGNER,   H.     (1)   The  Viscosity  of  Binary  Liquid   Mixtures.     Medd.   K. 

Vetenskapsakad.  Nobelinst.  3,  1  #22,  (1918);  13  pp.;  (2)  The  Viscosity 

and  Flocculation  of  Coarse  Suspensions.     Do.,  4,  #4,  (1919);  27  pp. 
EHRENBERG,   P.     Plasticity,   especially  of  Barium  Sulphate.     Z.   Angew. 

Chem.  24,  1957  (1911);  2  pp. 
EICHWALD,  E.     209,  Neuere  Untersuchungen  iiber  die  fliissigen  Kristalle. 

Diss.  Marburg  (1905);  39  pp.     (Phenol-water  mixtures.) 
EINSTEIN,  A.     188,  190,  206,  Eine  neue  Bestimmung  der  Molekulardimen- 

sionen.     Ann.  Physik.  19,  289  (1906);  18  pp.;  Ann.  Physik.   (4)  17, 

549  (1905);  1  p. 
VON   EISLER,    M.    &   LAUB,    M.     Viscosity   Estimations   in    Tuberculosis. 

Wien.  klin.  Wochschr.  25,  735  (1912);  5  pp. 
ELIE,    B.     83,  Variation  du  coefficient  de  viscosite  avec  la  vitesse.     J. 

physique  (2)  1,  224  (1882);  2  pp. 
ELLIS,  R.     Properties  of  Oil  Emulsions.     II.  Stability  and  Size  of  Globules. 

Z.  physik.  Chem.  80,  597  (1913);  20  pp. 
ELLIS,  R.  L.     Relation  between  Temperature  and  Viscosity  of  Lubricants. 

Met.  Chem.  Eng.  10,  546  (1912);  2  pp. 


368  INDEX 

ELSEY,  H.  M.  Conductivity  and  Viscosity  of  Solutions  in  Dimethyl- 
amine,  Trimethylamine,  Ethylamine,  Diethylamine,  Triethylamine, 
and  Propylamine.  J.  Am.  Chem.  Soc.  42,  2454  (1920);  23  pp. 

EMDEN,  R.  (1)  ITber  die  Ausstromungserscheinungen  permanenter  Gase. 
Habilitationsschrift,  Tech.  Hochschule  Miinchen,  Earth.  Leipzig 
(1899);  96  pp.;  Ann.  Phys.  Chem.  69,  426  (1899);  28  pp. 

EMLEY,  W.  E.  281,  282,  (1)  Measurement  of  Plasticity  of  Mortars  and 
Plasters.  Tech.  Paper  U.  S.  Bur.  Stands.  No.  169  (1920);  27  pp.; 
Trans.  Am.  Ceram.  Soc.  19,  523  (1917);  11  pp.;  (2)  The  Compressive 
Method  of  Measuring  Plasticity.  Proc.  Nat.  Lime  Mfgrs.  Assoc.  14, 
13  (1916);  8  pp.;  (3)  The  Clark  Viscometer.  Trans.  Am.  Ceram.  Soc. 
15,  401  (1913);  11  pp. 

EMO,  A.  Einfluss  der  Temperatur  auf  den  Reibungscoefficienten  des 
Quecksilbers  in  Capillarrohren.  Wied.  Beibl.  6,  730  (1882);  2  pp.; 
Estratto  dal  tesi  di  laurea  presentata  alia  Facolta  di  Scienze  fisico- 
mathematiche  della.  R.  Universita  di  Torino  nel  giugno.  (1881); 
26pp. 

ENGELMANN,  F.  &  ELPERS,  L.  Blood  Viscosity  in  Eclampsia  &  other  Dis- 
eases of  Women.  Gynakol.  Rundsch.  7,  319  (1913);  4  pp. 

ENGLER,  C.  Ein  Apparat  zur  Bestimmung  der  sogenannten  Viscositat 
der  Schmierole.  Chem.  Ztg.  9,  189  (1885);  2  pp.  Z.  Ver.  deut.  Ing. 
29,  882  (1885). 

ENGLER,  C.  &  KUNKLER,  A.  Viscosimeter  zur  Prufung  von  Oelen  bei 
constanter  Temperatur.  Dingler's  Polytech.  J.  276,  42  (1890);  6  pp. 

ENGLER.     3,  7,  324. 

ENGLISH,  S.  &  TURNER,  W.  The  Viscosities  of  Mixtures  of  Formamide 
with  the  Alcohols.  J.  Chem.  Soc.  170,  1656  (1913);  3  pp. 

ERCOLINI,  G.  35,  Attrito  interno  dei  liquidi  isolanti  in  campo  elettrico. 
Cimento  (5)  5,  249  (1903);  9  pp. 

ESPY,  W.  E.  Practical  Method  for  Calculating  Viscosity  of  Hydrocarbon 
Oil  Mixtures.  Petroleum  8,  No.  1,  27  (1919);  1  p. 

EUCKEN,  A.  252,  (1)  Physik.  Z.  12,  1101  (1901);  (2)  The  Thermal  Con- 
ductivity, the  Specific  Heat  and  the  Viscosity  of  Gases.  Physik.  Z. 
14,324  (1913);  8  pp. 

EULER,  H.  tlber  die  innere  Reibung  elektrolytische  Losungen.  Z.  physik. 
Chem.  25,  536  (1898);  6  pp. 

EULER,  L.  (1)  Tentamen  theoriae  de  frictione  fluidorum.  Novi  Comnien- 
tarii  Acad.  Scient.  Imper.  Petropolitani  6,  338(1756-7);  51pp.; 
(2)  Sectio  tertio  de  motu  fluidorum  lineari  potissimum  aquae.  Novi 
Commentarii  Acad.  Scient.  Imper.  Petropolitani  15,  219  (1770);  145 
pp. 

EUSTICE,  J.  Experiments  on  Stream  Line  Motion  in  Curved  Pipes.  Proc. 
Roy.  Soc.  (London)  85A,  119  (1911). 

EWALD,  C.  285,  tjber  die  Transpiration  des  Blutes.  Arch.  f.  Anat.  u. 
Physiol.  208  (1877);  32  pp.  Cp.  do  636  (1878);  2  pp. 

EWING,  J.  On  Hysteresis  in  the  Relation  of  Strain  to  Stress.  Report 
British  Assoc.*  502  (1889);  3  pp. 


INDEX  369 

EYTELWEIN.  1,  Untersuchungen  iiber  die  Bewegung  des  Wassers. 
Abhandl.  der  konigl.  Akad.  d.  Wissench.  Berlin  15,  137  (1814). 

FABRICANT-GOKUN,  T.  tlber  den  Einfluss  einiger  Salze  auf  die  zeitlichen 
Veranderungen  der  Viscositat  von  Colloiden.  Diss.  Zurich  (1910); 
14  pp.;  Z.  Chem.  Ind.  Kolloide  3,  #2  (1908). 

FABRY,  C.  &  PEROT,  A.  (1)  Measurement  of  Coefficient  of  Viscosity  of  Air. 
Nature  66,  383  (1896-7);  Abst.;  (2)  Sur  une  nouvelle  mesure  du  coeffi- 
cient de  viscosite  de  1'air.  Compt.  rend.  124,  281  (1897);  3  pp.;  Nature 
66,  383  (1896-7). 

FAIRBANKS,  F.  L.  (1)  The  Lubrication  of  Bearings  and  Cylinders.  Power 
42,  805  (1915);  4pp.;  (2)  Value  of  Cylinder  Oils.  Power  43,  701 
(1916). 

FALEK,  O.  Apparatus  for  Determining  the  Viscosity  of  Liquids.  Chem. 
App.  1,  100  (1914);  Ip. 

FANO,  G.  and  Rossi,  G.  284,  (1)  Sulla  viscosita  del  siero  di  sangue  solo  o 
mescolato  con  varie  sostanze.  Arch,  di  Fisiol.  1,  492  (1904);  12  pp.; 
(2)  Sulle  condizioni  determinant!  il  comportamento  della  viscosita 
del  siero  sanguigno  e  di  soluzioni  gommose  per  1'aggiunta  de  cloruro 
di  sodio  o  di  glucosio.  Arch,  di  Fisiol.  1,  609  (1904);  5  pp.;  (3)  Sulla 
viscosita  del  siero  sanguigno  nelle  lesioni  sperimentali  dell'apparecchio 
tiro-paratiroideo.  Do.  1,  792  (1904).  Cp.  Rossi. 

FARAGHER,  W.  F.  A  Standard  Viscometer  and  Standard  Method  for  Deter- 
mining Viscosities.  Why  Not?  J.  Ind.  Eng.  Chem.  6,  167  (1913);  1  p. 

FARMACHIDIS,  C.  Viscosity  of  the  Blood  in  Acute  Experimental  Mercury, 
Lead,  and  Phosphorus  Poisoning.  Clin.  med.  ital.  61,  273  (1912); 
34pp. 

FARR,  C.  C.,  COLERIDGE,  C.  and  MACLEOD,  D.  B.  The  Viscosity  of  Sulphur. 
Proc.  Roy.  Soc.  London  A  97,  80  (1919);  18  pp. 

FARROW,  F.  D.  290,  291,  The  Viscosity  of  Aqueous  Solutions  of  Sodium 
Palmitate  and  the  Influence  of  Electrolytes  on  the  Same.  J.  Chem. 
Soc.  101,  347  (1912);  10  pp. 

FAUST,  O.  141,  (1)  Die  innere  Reibung  von  Fliissigkeitgemischen,  ihre 
Abhangigkeit  von  der  Temperatur,  und  die  Verwandschaft  der  inneren 
Reibung  von  Fliissigkeiten  mit  ihrem  Dampfdruck.  Z.  physik. 
Chem.  79,  97  (1912);  26  pp.;  (2)  Viscosity  of  Liquids  at  High  Pressures. 
Z.  physik.  Chem.  86,  479  (1914);  15  pp.;  Rep.  of  the  Lubricants  and 
Lubrication  Inquiry  Committee,  Dept.  of  Scientific  and  Industrial 
Research,  Advisory  Council,  Appendix  I.  London  (1920);  13  pp.;  (3) 
Viscosity  Measurements.  Z.  physik.  Chem.  93,  758  (1919);  3  pp. 

FAUST,  O.  &  TAMMANN,  G.  (1)  tlber  Verschiebungselastizitat  und  ihren 
Zusammenhang  mit  der  inneren  Reibung.  Z.  physik.  Chem.  71r  51 
(1910);  8  pp.  Cp.  also  Tammann;  (2)  Displacement  Elasticity  and 
its  Relation  to  Viscosity.  Z.  physik.  Chem.  71,  51  (1910);  8  pp. 

FAUSTEN,  A.     Versuche  zur  Bestimmung  einer  allgemeinen  Formel  zum 
Berechnen    der    Ausflussgeschwindigkeit   beim    Fliessen    des   Wassers 
durch  Rohren.     Diss.  Bonn  (1906). 
24 


370  INDEX 

FAWSITT,  C.  6,  (1)  Studies  in  Viscosity.  The  Electrochemist  and  Metal- 
lurgist 3,  664  (1904);  4  pp.;  Electrochemist  &  Metallurgist  3,  801 
(1904);  (2)  Physico-Chemical  Investigations  in  the  Amide  Group. 
Proc.  Roy.  Soc.  Edinburgh  25,  51  (1904);  9. pp.;  (3)  On  the  Deter- 
mination of  Viscosity  at  High  Temperatures.  Proc.  Roy.  Soc.  London 
80,  290  (1908);  9  pp.;  Proc.  Chem.  Soc.  24,  146  (1908);  1  p.;  J.  Chem. 
Soc.  93,  1299  (1908);  9  pp. 

FlNKENER.       17. 

FEA,  L.     Practical  Determination  of  the  Viscosity  of  Oils.     Proc.   Int. 

Assoc.  Testing  Materials  2  [13]  xx!2. 

FEILD,  A.  L.  (1)  Slag  Control  in  the  Iron  Blast  Furnace  by  Means  of 
Slag  Viscosity.  Chem.  Met.  Eng.  19, 294  (1918) ;  6  pp. ;  (2)  Slag  Viscosity 

Tables  for  Blast  Furnace  Work.     Bull.  Am.  Inst.  Mining  Eng.,  No. 

136,791  (1918);  3  pp. 
FIELD.  A.  &  ROYSTER,  P.     Temperature-Viscosity  Relations  in  the  Ternary 

Systems  CaO-AUOs-SiOz.     Bull.  Am.  Inst.  Mining  Eng.  2037  (1917); 

6  pp.;  U.  S.  Bureau  of  Mines  Tech.  Paper  187  (1918);  38  pp. 
VON  FELLENBERG,  T.     Method  for  Determining  Adulteration.     Mitt.  Leb- 

ensm.  Hyg.  2,  161  (1912);  17pp. 

FELS,  J.  (1)  tJber  die  Viscositat  des  Leimes,  ein  neues  Priifungsverfahren. 
Chem.  Ztg.  21,  56  (1897);  1  p.;  (2)  Do.,  Chem.  Ztg.  21,70  (1897);  1  p.; 

(3)  Do.,  Chem.  Ztg.  22,  376  (1898);  2  pp.;  (4)  Beitrage  zur  Bestimmung 

der  Viscositat  des  Leimes.     Chem.  Ztg.  25,  23  (1901);  1  p.;  (5)  Chem. 

Ztg.  24,  994  (1900). 
FELTZ  &  RITTER,  E.     De  1'action  des  sels  biliares  sur  le  pouls,  la  tension, 

la  respiration  et  la  temperature.     Compt.  rend.  82,  567  (1876). 
FENNEL,  L.     ttber  die  Bewegung  eines  festen  Korpers  in  einer  tropfbaren 

Flussigkeit.     Diss.  Marburg  (1886);  43  pp. 
FERRAI,    C.     Ricerche    viscosimetriche  sul    sangue    asfittico.     Arch,      di 

Fisiologia  1,  385  (1904);  Richerche  viscosimetriche  sul  sangue  asffitico. 

Arch,  fisiol.  1,  #4. 
FERRERO,   E.     Sull'   attrito  interno   nelle  soluzioni   di  alume  di  cromo. 

Cimento  (5)  1,  285  (1901);  3  pp. 
FICK,  A.     tJber  den  Druck  in  den  Blutcapillaren.     Pflugers  Arch.  42,  482 

(1888). 
FILIPPI,  E.     Relation  between  Changes  in  Viscosity  and  Surface  Tension 

of   the   Living    Blood   under   the  Action  of  Different  Medicines.     Lo 

Sper.  63,  373  (1909);  35  pp. 
FINDLAY,  A.     172,  The  Viscosity  of  Liquid  Mixtures  at  the  Temperatures 

of   their   Boiling-Points.     Report   British    Assoc.    1905,   365    (1906). 

Z.  Physik.  Chem.  69,  203  (1909);  15  pp. 
FISCHER,  E.  &  SCHNEIDATJER,  E.     tlber  das  Aufsteigen  von  Salzlosungen  in 

Filtrirpapier.     Lieb.  Ann.  272,  156  (1892);  13  pp. 
FISCHER,  F.     Uber  Untersuchungen  von  Schmierolen.     Dingler's  polytech. 

J.  236,  487  (1880);  10  pp. 
FISCHER,  R.     A  New  Viscometer.     Chem.  Ztg.  44,  622  (1920);  Z.  angew. 

Chem.  34,  Aufsatzteil,  153  (1920);  1  p. 


INDEX  371 


FISHER,   W.     50,  58,  59,   (1)   Temperature  coefficients  of  Gas 

IV.  Apparent  Relation  between  Viscosity  and  Specific  Heat.  Phys. 
Rev.  29,  147  (1909);  6  pp.;  (2)  The  Temperature  Coefficients  of  Gas 
Viscosity.  I.  Sutherland's  Equation,  Pure  Gases.  Phys.  Rev.  24, 
385  (1907);  16  pp.;  (3)  The  Constants  of  Gas  Viscosity.  Do.  24,  237 
(1907);  2  pp.;  (4)  The  Coefficients  of  Gas  Viscosity.  Phys.  Rev.  28, 
73  (1909);  33  pp.;  (5)  The  Kinetic  Pressure-drop  Correction  in  the 
Transfer  Method  for  Gas  Viscosity.  Phys.  Rev.  32,  216  (1911);  3  pp. 

FISCHLER,  J.  Molecular  Conductivity  and  Inner  Friction  of  Mixtures  of 
Alcohol  or  Acetone  with  Benzene  and  Nitrobenzene.  Z.  Elektrochem. 
19,  126  (1913);  6  pp. 

FITZGERALD,  F.  (1)  The  Viscosity  of  Liquid  Ammonia,  Sulfur  Dioxide, 
and  Methyl  Amine.  Proc.  J.  Am.  Chem.  Soc.  32,  104  (1910);  (2) 
Electrical  Conductivity  of  Solutions  in  Methylamine  and  Ethylamine; 
Fluidity  of  Ammonia  and  Sulfur  Dioxide  and  the  Fluidity  of  Certain 
Solutions  in  these  solvents.  J.  Phys.  Chem.  16,  621  (1912);  40  pp. 

FITZGERALD,  G.  F.  Turbulent  Motion  in  Viscous  Liquids.  Trans.  Roy. 
Sci.  Soc.  Dublin  6,  289  (1888-90). 

FLOWERS,  A.  E.  Viscosity  Measurement  and  a  New  Viscometer.  Proc. 
Am.  Soc.  Testing  Materials  14,  II,  568  (1914);  49  pp. 

FOL,  J.  A.  (1)  Relation  between  the  Amount  of  Resins  and  the  Viscosity  of 
Rubber  Solutions.  Orig.  Com.  8th  Intern.  Congr.  Appl.  Chem. 
9,  71  (1912);  2  pp.;  (2)  Determination  of  the  Viscosity  of  Rubber  Solu- 
tions. Caoutchouc  and  Gutta-percha  10,  6973,  7059  (1913);  (3)  The 
Relation  between  the  Resin  Content  and  the  Viscosity  of  India  Rubber 
Solutions.  Gummi  Ztg.  7,  247  (1912);  (4)  Viscosity  Measurements 
of  India  Rubber  Solutions.  Kolloid-Z.  12,  131  (1913). 

FORBES.     Viscosity  Theory  of  Glacial  Motion. 

FORCH,  C.  Oberflachenspannung  und  Reibungskoefficient  flussiger  Luft. 
Physic.  Z.  1,  177  (1900). 

FORCHHEIMER,  P.  L.     Hvdraulik.  Teubner,  Berlin  (1914);  566  pp. 

FOUGUET  DE  NASSANDRES.  Viscosite  des  sirops.  Bui.  Ass.  Chimistes  22, 
1186  (1905). 

FotrssEREAU,  G.  6,  183,  193,  194,  (1)  Recherches  experimentales  sur  la 
resistance  electrique  des  substances  isolantes.  1.  Relation  de  la  resis- 
tance avec  le  frottement  interieur  de  1'eau.  Ann.  chim.  phys.  (6) 
5,  348  (1885);  (2)  Do.  2.  Coefficients  de  frottement  interieur  des  sels 
fondus.  Ann.  chim.  phys.  (6)  5,  359  (1885). 

FRANK,  J.  Physical  Properties  of  Colloidal  Solutions.  Kolloidchem.  Bei- 
hefte4,  195  (1913);  33  pp. 

FRANKENHEIM,  M.  Krystallization  und  Amorphie.  J.  Prakt.  Chem.  64, 
433  (1851). 

FREUND,  G.  A.  Aeris  motu  circa  cylindrum  qui  rotatur.  Inaug.  diss. 
Berlin  (1862);  17pp. 

FREUNDLICH,  H.  The  difference  of  potential  acting  in  electroosmosis  and 
related  phenomena.  Kolloid-Z.  28,  240  (1921);  2  pp. 

FREUNDLICH,  H.  &  ISHIZAKE,  C.     (1)  Die  Koagulationgeschwindigkeit  von 


372  INDEX 

Al  (OH)s-Solen,  gemessen  an  der  Aenderung  ihrer  Zahigkeit.     Trans. 

Faraday  Soc.  9,  (1913);  14  pp.;  Z.  Chem.  Ind.  Kollide  12,  230  (1913); 

8pp. 

FREY,  V.     The  Viscosity  of  the  Blood.     Transvaal  Medical  J.  April  (1908). 
FRIEDLANDER,   J.     7,   94,  102,  Uber  merkwurdige  Erscheinungen  in  der 

Umgebung  des  kritischen  Punktes  teilweise  mischbarer  Fliissigkeiten. 

Z.  physik.  Chem.  38,  399  (1901);  56  pp. 
FRIEDLANDER,  J.  &  TAMMANN,  G.     Ueber  die  Krystallisationsgeschwindig- 

keit.     Z.  physik.  Chem.  24,  152  (1897);  8  pp. 
FRIG.     Compt.  rend.  154,  31  (1912). 
FRITZSCHE.     Untersuchungen  iiber  den  Stromungswiderstand  der  Gase  in 

geraden     zylindrischen      Rohrleitungen.     Mitt.      Forschungsarbeiten 

Verein  Deutscher  Ingenieure,  Heft  60,  Springer,  Berlin  (1908);  71  pp.; 

Abst.  Zeitschr.  Ver.  deutsch  Ing.  81  (1908). 

FRUMPP.     The  Viscosity,  the  Haemoglobin  and  Protein  Content  of  Chil- 
dren's Blood.     Munch.  Med.  Wochschr.  66,  2145  (1910). 
FUETH  &  KROENIG.     Centralblatt  f.  Gynakologie  701  (1901). 
FULD,  E.     286,  Uber  die  Milchgerinnung  durch  Lab.     Hofmeister's  Beitr. 

z.  Chem.  Physiol.  Path.  2,  169. 

GAEDE,  W.  (1)  External  Friction  in  Gases.  Ann.  Physik.  41,  289  (1913); 
47  pp.;  8,  277;  (2)  Gas  Friction  and  a  New  Principle  for  Air  Pumps. 
Electrician  70,  48  (1912);  2  pp. 

GALDI,  F.  (1)  Contribution  to  the  Study  of  the  Relation  between  Viscosity, 
Specific  Gravity  and  Pressure.  II.  Tammasi  3,  No.  7;  (2)  Influence  of 
Pressure  and  Specific  Gravity  upon  the  Relative  Viscometric  Coeffi- 
cient of  Organic  Liquids  as  Compared  with  Solutions  of  Crystalloids 
and  Colloids.  Giorn.  ind.  sci.  med.  (1909);  (3)  Relation  between 
Time  of  Outflow  and  Pressure.  Riv.  chim.  microscop.  clinic  9,  (1909). 

GALILEO.     1,  Works  2,  537  (1718). 

GARDNER,  H.  A.  and  HOLDT,  P.  C.  The  Measurement  of  the  Consistency 
of  Varnish.  Paint  Mnfrs.  Assoc.  of  the  U.  S.  (Sci.  Sect.)  Circular  127 
(1921);  51  pp. 

GARNIER,  G.  Sur  la  fluidity  du  nickel  fondu.  Compt.  rend.  124,  1447 
(1897);  2  pp. 

GARRETT,  H.  212,  213,  The  Viscosity  and  Composition  of  some  Colloidal 
Solutions.  Diss.  Heidelburg  (1903);  Phil.  Mag.  (6)  6,  374  (1903); 
4pp. 

GARTENMEISTER,  R.  2,  7,  17,  Die  Zahigkeit  flussiger  Kohlenstoffverbind- 
ungen  und  ihre  Beziehung  zur  chemischen  Konstitution.  Z.  physik. 
Chem.  6,  524  (1890);  27pp. 

GARVANOFP,  J.  tJber  die  innere  Reibung  in  Olen  und  deren  Anderung  mit 
der  Temperatur.  Wien.  Sitzungsber  (2A)  106,  873  (1894);  14  pp. 

GAUNT,  R.  Viscosity  of  Rubber  Solutions.  J.  Soc.  Chem.  Ind.  33,  446 
(1913);  6pp.  Cp.  India  Rubber  J.  47, 1045-1093  (1913);  6  pp. 

GAZARIAN,  G.  (1)  A  General  Relation  between  the  Physical  Properties  of  a 
Substance;  Application  to  Viscosity,  Surface  Tension,  Heat  of  Vapori- 


INDEX  373 

zation,  etc.     Compt.  rend.  153,  1071  (1912);  (2)  A  General  Relation 

Between  the  Physical  Properties  of  a  Substance:  Applic.  to  Densities. 

Compt.  rend.  153,  871  (1912);  3  pp. 
GAZETTI,  C.     The  Influence  of  Alkali  Salts  on  the  Viscosity  of  Proteins. 

Arch,  di  Fis.  11,  173  (1913). 
GEOFFROY,  L.     Sur  les  resistances  qu'eprouve  une  surface  mobile  de.  la 

part  d'un  milieu  fluide  dans  lequel  elle  se  meut.     Ann.  de  1'ecole  norm. 

(2)  7,  215  (1878);  12  pp.;  Compt.  rend.  88,  573  (1879). 
GERNEZ,  D.     Recherches  sur  la  dur6e  de  la  solidification  des  corps  sur- 

fondus.     Ann.  de  1'ecole  norm.  (3)  1,  239  (1884);  44  pp.;  Journ.  de 

Phys.  (2)  2,  159  (1883);  4  pp. 
GERSTNER.     1,   6,   127,   Versuche   iiber  die   Fliissigkiet   des   Wassers  bei 

verschiedenen  Temperaturen.     Gilbert's  Annalen  5,  160  (1800);  24  pp. 

Cp.  Abhandl.  der  konigl.  Bohmischen  Gesellschaft  der  Wissenschaften, 

Phys.-math.  Teil  3,  141  (1798);  19  pp. 
GESSNER.     Starke-Viscose    und    Alkalistarke-Zanthogenate.     Diss.    Tech. 

Hochsch.  Hannover  (1910);  48  pp. 
GETMAN,  F.     186,  (1)  J.  chim.  phys.  4,  386  (1906);  20  pp.;  (2)  J.  chim. 

phys.  6,  344  (1907);  19  pp.;  (3)  A  Study  of  the  .Solutions  of  Some 

Salts  Exhibiting  Negative  Viscosity.     J.  Am.  Chem.  Soc.  30,  721  (1908); 

16  pp.;   (4)   The  Viscosity  of  Non- Aqueous  Solutions  of  Potassium 

Iodide.     J.  Am.  Chem.  Soc.  30,  1077  (1908);  8  pp.;  (5)  Viscosite"  et 

volume  ionique.     J.  chim.  phys.  6,  577  (1908);  6  pp.;  Cp.  Graham. 
GIBSON,  A.     170,  171,  (1)  An  Investigation  of  the  Resistance  to  the  Flow  of 

Air  through  a  Pipe,  with  the  Deduction  and  Verification  of  a  Rational 

Formula.     Phil.  Mag.  (6)  17,  395  (1909);  14  pp.;  (2)  Conversion  of 

kinetic  energy  to  pressure  energy  in  the  flow  of  water  through  passages 

having    divergent   boundaries.     Engineering    93,    205    (1912);    2    pp. 

Cp.  Grindley. 
GIBSON,  W.     The  Influence  of  Volume  Change  on  the  Fluidity  of  Mixtures 

of  Miscible  Liquids.     Phil.  Mag.  (6)  27,  662  (1914);  7  pp. 
GIBSON,  W.  H.     Viscosity  of  solutions  of  cellulose.     J.  Chem.  Soc.  117,  479 

(1920);  16pp. 
GIBSON,  W.  H.  and  JACOBS,  L.  M.     The  Falling-Sphere  Viscometer.     J. 

Chem.  Soc.  117,  473  (1920). 
GIBSON,  W.  &  McCALL,  R.     The  Viscosity  of  Solutions  of  Nitrocellulose 

in  Ether-Alcohol.     J.  Soc.  Chem.  Ind.  39,  172  T. 
GILCHRIST,  L.     242,  An  Absolute  Determination  of  the  Viscosity  of  the  Air. 

Physic.  Rev.  II 1,  124  (1913);  Physik.  Z.  14,  160  (1913);  5  pp. 
GILL,  A.  H.     (1)  Drugs,  Oils  &  Paints  26,  #56-7;  (2)  Oil  Analysis,  Lippin- 

cott  &  Co. 
GIRARD.     1,  6,  30,  Sur  le  mouvement  des  fluides  dans  les  tubes  capillaires  et 

1'influence  de  la  temperature  sur  ce  mouvement.     Mem.  de  la  Classe 

des  Scienc,  Math,  et  Phys.  de  1'Inst.  de  France  14,  249  (1813,  1814, 

1815);  Ann.  de  Chim.  (2)  16,  129  (1821);  Sur  I'ecoulement  lineaire  de 

diverses  substances  liquides  par  des  tubes  capillaires  de  verre.     Mem. 

de  1'acad.  roy.  des.  Sci.  de  1'Inst.  de  France  1,  187  (1816);  74,  pp. 


374  INDEX 

Cp.  Extract  of  same  Ann.  Chim.  phys.  (2)  4,  146  (1817);  19  pp.     Also 

Am.  Chim.  phys.  (2)  1,  436  (1816);  8  pp.     Sur  1'ecoulement  de  1'ether 

et  quelques  autres  fluides  par  des  tubes  capillaires  de  verre.     Do.  1, 

260  (1816);  14pp. 

GIRARD,   M.   and  HENRY,  V.   C.     Etudes  sur  1' agglutination.     I.   Agglu- 
tination des  globules  rouges  par  les  colloides.     Compt.  rend.  Soc.  Biol. 

66,  866,  931,  974  (1903);  67,  34,  65  (1904). 
GIRAULT,  C.     De  la  resistance  de  1'air  dans  le  mouvement  oscillatoire  du 

pendule.     Me'moires    de   1' Academic   imperiale    des   sciences,    arts   et 

belles-lettres  de  Caen  1  (1860);  45  pp. 
GLASER.     62,  53,  54,  239,  240,  Uber  die  innere  Reibung  zaher  und  plastisch- 

fester  Korper  und  die  Gliltigkeit  des    Poiseuillesche    Gesetz.     Diss. 

Erlangen  (1906);  Erlanger  Berichte  38,  147  (1906);  4  pp.;  Ann.  Physik. 

(4)  22,  694  (1907);  26pp. 
GLAUBERMANN,  J.     Influence  of  Pressure  on  the  Viscosity  Coefficient  of 

the  Blood.     Berl.  Klin.  Wochschr.  49,  1991  (1913);  2  pp. 
GOKTJM.     (1)  tJber  die  Beeinflussung  der  Viskositat  der  Kolloide  durch 

Elektrolyten.     Z.  Chem.  Ind.  Kolloide  3,  84  (1907);  (2)  Viscosity  of 

Gelatine.     Z.  Ind.  Kolloide  3,  84  (1908). 
GOLDSCHMIDT,    F.     Viscosity    of    Soap    Solutions.     Seifensieder    Ztg.    41, 

337  (1913);  1  p. 
GOLDSCHMIDT,  F.  &  WEISMAN,  L.     Aqueous  Solutions  of  Ammonia  Soaps. 

Z.  Elektrochem.  18,  380  (1912);  Kolloid-Z.  12,  18  (1913);  Seifensieder 

Ztg.  41,  337  (1914). 
GOODWIN;  H.  M.  and  KALMUS,  H.  T.     On  the  Conductance  and  Fluidity 

of  Fused  Salts.     Phys.  Rev.  27,  322  (1908);  6  pp.     Cp.  Lorenz  and 

Kalmus. 
GOODWIN,  H.  and  MAILEY,  R.     (1)  On  the  Density,  Electrical  Conductivity, 

and  Viscosity  of  Fused  Salts  and  their  Mixtures.     Phys.  Rev.  25,  469 

(1907);  21  pp.;  (2)  Do.,  Do.  26,  28  (1908);  33  pp. 
GORKE,   H.     179,  et  seq.,   tlber  Losungen  stark  dissocierter  Electrolyte. 

Diss.  Leipzig  (1915);  49  pp.     (Densities,  conductivities,  and  viscosities 

of  aqueous  solutions  of  NH4CNS,    KCNS,   LiCl,   NH4NOS,   KNO3, 

NaNOs,  AgNOs,  &  KI). 
GOSTUNIN,   M.   P.   &  LEDANTU,   P.   A.     Contributions  to  the  Study  of 

Substances  having  Large  Coefficients  of  Viscosity.     J.   Russ.   Phys. 

Chem.  Soc.  (Phys.  Pt.)  44,  241  (1913);  11  pp.     Cp.  Vienberg. 
GRAETZ,  L.     128, 129,  134,  (1)  Schlomilchs  Zeitsch.  f.  Math.  26,  316  (1879); 

(2)  t)ber  die  Reibung  von  Flussigkeiten.     Wied.  Am.  34,  25  (1888); 

15pp.;  (3)  Reibung.  Handb.  d.  Phys.  Breslau  1,  595  (1890);  30  pp.; 

(4)  tJber  die  Warmeleitungsfahigkeit  von  Flussigkeiten.     Wied.  Am. 

26,  337  (1885);  21  pp.;  Continuation  of  Wied.  Am.  18,  7  (1883);  (5) 

Winkelman's  Handbuch  d.  phys.  I,  1373  (1910);  37  pp. 
GRAHAM,  M.     A  Study  of  the  Change  from  Violet  to  Green  in  Solutions 

of  Chromium  Sulphate.     Am.  Chem.  J.  48,  145  (1912);  45  pp.     Cp. 

Getman. 
GRAHAM,  T.     2,  79,  106,  198,  213,  241,  251,  (1)  On  the  motion  of  Gases. 


INDEX  375 

Phil.  Trans.  Lond.  136,  573  (1846);  59  pp.;  (2)  On  the  Motion  of  Gases 
Part  II.  Phil.  Trans.  Lond.  139,  349  (1849);  (3)  tfber  die  Bewegung 
der  Gase.  Lieb.  Ann.  76,  138  (1850);  13  pp.;  (4)  On  Liquid  Trans- 
piration in  relation  to  Chemical  Composition.  Phil.  Trans.  161, 
373  (1861);  (5)  On  Liquid  Transpiration  in  relation  to  Chemical  Com- 
position (abstract  of  3).  Phil.  Mag.  (4)  24,  238  (1862);  3  pp.; 
(6)  Liquid  Diffusion  applied  to  Analysis.  Phil.  Trans.  161,  183  (1861). 

'SGRAVESANDE.  1,  Philosophiae  Newtoniae  Institutiones.  1st  Ed.  (1719); 
4th  Ed.  (1748). 

GRAWITZ,  E.  (1)  Klinisch-experimentelle  Blutuntersuchungen.  Z.  f.  klin. 
Med.  21,  459  (1892);  16  pp.;  (2)  Do.  22,  411  (1893);  38  pp. 

GRAY,  A.  On  the, Relation  between  Temperature  and  Internal  Viscosities 
of  Solids.  The  Electrician  22,  838  (1901);  British  Assoc.,  Glasgow 
629  (1901). 

GRAY,  A.,  BLYTH,  V.  and  DUNLOP,  J.  On  the  Effects  of  Changes  of  Tempera- 
ture on  the  Elasticities  and  Internal  Viscosities  of  Metal  Wires.  Proc. 
Roy.  Soc.  London  67,  180  (1900);  18  pp. 

GRAY,  A.  and  WOOD,  A.  On  the  Effect  of  a  Longitudinal  Magnetic  Field 
on  the  Internal  Viscosity  of  Wires  of  Nickel  and  Iron,  as  shown  by 
Change  of  the  Rate  of  Subsidence  of  Torsional  oscillations.  Proc. 
Roy.  Soc.  70,  294  (1902);  9  pp. 

GRAY,  T.  T.  A  comparison  of  the  Engler  and  Saybolt  Viscosities  of  Mixed 
Oils.  Orig.  Com.  8th  Intern.  Congr.  Appl.  Chem.  10,  153  (1912);  5 
pp. 

GREEN,  HENRY.  231,  266,  Further  Development  of  the  Plastometer  and 
Its  Practical  Application  to  Research  and  Routine  Problems.  Proc. 
Am.  Soc.  Testing  Materials,  II,  20,  (1920);  44  pp.  Cp.  Bingham  and 
Green. 

GREEN,  W.  H.  195,  Studies  on  the  Viscosity  and  Conductivities  of  some 
Aqueous  Solutions.  Part  I.  J.  Chem.  Soc.  93,  2023  (1909);  26  pp.; 
Part  II.  J.  Chem.  Soc.  93,  2049  (1909);  15  pp.  Cp.  Proc.  Chem.  Soc. 
24,  187  (1909). 

GREINER,  E.  tlber  die  Abhangigkeit  der  Viscositat  in  Silikatschmelze 
von  ihrer  chemischen  Zusammensetzung.  Inaug.  Diss.  Zena  (1907); 
57pp. 

GRIFFITHS.  197,  On  the  Viscosity  and  Electrolytic  Resistance  of  a  Gelatine 
Solution.  Proc.  Manchester  Lit.  &  Phil.  Soc.  41,  IX  (1896);  2  pp. 

GRIFFITHS,  A.  and  GRIFFITHS,  MRS.  CONSTANCE.  Viscosity  of  Water  at 
Low  Rates  of  Shear.  Proc.  Phys.  Soc.  (London)  33,  231  (1921);  11  pp. 

GRIFFITHS,  A.  and  KNOWLES,  Miss  C.  H.  The  Resistance  to  the  Flow  of 
Water  along  a  Capillary  Soda-Glass  Tube  at  Low  Rates  of  Shear. 
Proc.  Phys.  Soc.  (London)  24,  350  (1912). 

GRINAHOVSKII,  K.  P.  Cause  of  the  Abnormal  Linear  Velocity  of  Crystal- 
lization of  Supercooled  Crystalline  Substances.  J.  Russ.  Phys.  Chem. 
Soc.  45,  1210  (1914);  38  pp. 

GRINDLEY,  J.  H.  and  GIBSON,  A.  H.     51,  242,  On  the  Frictional  Resistance 


376  INDEX 

to  the  Flow  of  Air  through  a  Pipe.  Proc.  Roy.  Soc.  London  80A,  114 
(1908);  26  pp.  Cp.  Gibson. 

GRONATJ.  tlber  die  Bewegung  schwingender  Korper  im  widerstehenden 
Mittel.  Programm  Danzig  (1850). 

GROSSMANN,  L.  192,  (1)  Theorie  der  numerischen  Berechnung  der  Con- 
stanten  der  inneren  Reibung  und  ausseren  Rcibung  von  Gasen  und 
Flussigkeiten  mittelst  schwingender  Scheiben,  nebst  experimenteller 
Untersuchung  der  ausseren  Reibung  zwischen  Wasser  und  Quecksilber. 
Diss.  Breslau  (1880);  56  pp.;  (2)  tlber  die  Bestimmung  der  inneren 
Reibungsconstanten  von  Gasen  und  Flussigkeiten  mittelst  schwing- 
enden  Scheiben.  Wied.  Ann.  16,  619  (1882);  15  pp.,  Abstr.  of  1; 
(3)  Das  Product  innerer  Reibung  und  galvanischer  Leitung  der  Flus- 
sigkeiten in  Bezug  auf  die  Temperatur.  Wied.  Ann.  18,  119  (1883);  17 
pp. 

GROTIAN,  O.  6,  127,  192,  (1)  Die  Reibungskonstanten  einiger  Salzlosungen 
und  ihre  Beziehungen  zum  galvanischen  Leitungsvermogen.  Pogg. 
Ann.  157,  130,  237  (1876);  17  pp.  and  21  pp.;  (2)  Weitere  Mitteilungen 
iiber  den  Zusammenhang  zwischen  der  Viscositat  und  dem  galvanischen 
Leitvermogen  verschiedener  Flussigkeiten.  Pogg.  Ann.  160,  238 
(1877);  (3)  Analogien  zwischen  der  Fluiditat  und  dem  galvanischen 
Leitungsvermogen.  Wied.  Ann.  8,  529  (1879). 

GROUT,  F.  F.  (1)  Clays,  Limestones  and  Cements.  W.  Va.  Geol.  Surv. 
3,  35  (1906);  20  pp.;  (2)  The  Plasticity  of  Clays.  J.  Am.  Chem.  Soc.  17, 
1037  (1895);  12pp. 

GROUT,  F.  F.  and  POPPE,  F.  The  Plasticity  of  Clay.  Trans.  Am.  Ceram. 
Soc.  14,  71  (1912);  11  pp. 

GRUMMEL,  E.  Observations  compares  entre  la  vitesse  des  reactions  et 
la  fluidit^  du  milieu.  J.  chim  physique  9,  143  (1911);  17  pp. 

GRUNEISEN,  E.  7,  (1)  Bewegung  tropfbarer  Flussigkeiten  durch  gerade 
und  gewundene  Kapillaren.  Wiss.  Abh.  phys.  Reichsanst  4,  151 
(1905);  29  pp.;  (2)  Innere  Reibung  wasserigen  Salzlosungen  und  ihren 
Zusammenhang  mit  der  Elektrolytischen  Leitung.  Wiss.  Abh.  phys. 
Reichanst.  4,  237  (1905);  29  pp. 

GRUNMACH,  L.  (1)  Determination  of  the  Surface  Tension  and  other  Physical 
Constants  of  Mixtures  of  Acetic  Acid  and  Water.  Wiss.  Abhand.  d. 
Kaiserl.  Normal  Eichungskom.  7,  45-130;  Ann.  Phys.  28,  217,  41  pp.; 
(2)  Einfluss  der  Zahigkeit  auf  die  Kapillarkonstant  bei  Essigsaure- 
Wassermischungen.  Festshrift  Boltzmann  460  (1904). 

GUNTHER,  P.  The  Viscosity  of  Hydrogen  at  Low  Temperatures.  Sitz. 
preuss.  Akad  720  (1920);  9  pp. 

GUEROUT,  A.  2,  107,  (1)  Sur  le  coefficient  d'£coulement  capillaire.  Compt. 
rend.  81,  1025  (1875);  2  pp.;  (2)  Recherches  sur  le  coefficient  d'6coule- 
ment  capillaire.  Compt.  rend.  83,  1291  (1876);  3  pp.;  (3)  Influence  de 
la  temperature  sur  le  coefficient  d'6coulement  capillaire  des  liquides. 
Compt.  rend.  79,  1201  (1874). 

GULBRING,  A.     Significance  of  Leucocytes  in  Connection  with  Viscosity 


INDEX  377 

of  the  Blood.     Inaug.  Diss.  Stockholm  (1913);  140  pp.,  Hygiea  75; 

Zentr.  Biochem.  Biophys.  16,  518  (1913). 
GUMBEL.     Der     Widerstand     geschmierter    Flachen.     Monatsblatter    des 

Berliner  Bezirksvereines  deutscher  Ingenieure.  Julius  Springer.  Berlin 

(1914);  87. 
GUNZBURG,  I.     The  Influence  of  Uranium  and  Potassium  on  the  Viscosity 

of  Liquid  Colloids.     Arch.  Neerl.  Physiol.  4,  233  (1920);  10  pp. 
GURNBY,   H.   P.     A  Method  of  Measuring  Absolute  Viscosity.     J.   Am. 

Chem.  Soc.  34,  24  (1912);  4  pp.;  Mat.  Grasses  5,  2614  (1912). 
GURNEY,  L.     256,  (1)  The  Viscosity  of  Water  at  Very  Low  Rates  of  Share. 

Phys.  Rev.  26,  98  (1908);  22  pp.;  (2)  Effects  of  the  Soluble  Constitu- 
ents  of   Glass  upon   the   Viscosity    of  Water  at  Very  Low  Rates  of 

Shear.     Phys.  Rev.  26,  123  (1908);  1  p. 
GUTHE,  K.     Some  Cases  of  Excessive  Damping  of  Torsional  Vibrations. 

Phys.  Rev.  26,  201  (1908);  Science  (N.  S.)  27,  572  (1908);  Iowa  Acad. 

Sciences  147  (1908). 
GUTHRIE,  F.     On  the  Influence  of  Temperature  on  the  Passage  of  Air 

through  Capillary  Tubes.     Phil.  Mag.  (5)  5,  433  (1878);  7  pp. 
GUTZEIT.     286,  Uber  Anderungen  in  der  physikalischen  Beschaffenheit  der 

Milch  unter  Einwirkung  der  Labflussigkeit.     Milchztg.  24,  745. 
GUY,  J.  S.  and  JONES,  H.  C.     Conductivity  and  Viscosity  in  Mixed  Solvents 

Containing  Glycerol.     Am.  Chem.  J.  46,  131  (1911);  67  pp. 
GUYE,  C.  E.     (1)  Internal  Friction  of  Solids;  Its  V  ariation  with  the  Tempera- 
ture.    Arch.   sci.   phys.   nat.  24,  535  (1913);  7  pp.;  (2)  The  Interior 

Viscosity  of  Solids;  Its  Variation  with  the  Temperature.     J.  physique 

(5)  2,  620  (1913);  26  pp.;  (3)  Viscosity  of  Metals  as  a  Function  of  the 

Temperature.     Arch.  sci.  phys.  nat.  29,  474  (1911). 
GUYE,  C.  E.  and  BARBIER,  P.     Remark  on  the  Internal  Friction  of  Quartz 

Filaments  at  Low  Temperatures.     Arch.  sci.  phys.  nat.  46,  326  (1918); 

3pp. 
GUYE,    C.    E.    and   EINHORN-BODZECHOWSKI.     The   internal   Friction    of 

Quartz  Fibres  at  Low  Temperatures.     Arch.  sci.  phys.  nat.  41,  376 

(1916);  25  pp. 
GUYE,  C.  &  FREEDERICKSZ,  V.     (1)  Internal  Friction  of  Solids  at  Low 

Temperatures.     Compt.    rend.   149,   1066    (1909);   3  pp.;   Arch.   sci. 

phys.  nat.  29,  49  (1909);  (2)  Internal  Friction  of  Metals  at  Low  Temps. 

Arch.  Sci.  phys.  nat.  29,  157  (1909);  18  pp.;  (3)  Do.  29,  261  (1909); 

29pp. 
GUYE,  C.  E.  and  MINTZ,  S.     1,  (1)  Etude  sur  la  viscositS  de  quelques 

mStaux  en  fonction  de  la  temperature.     Arch.  sci.  phys.  nat.  (4)  26, 

136  (1908);  31  pp.;  (2)  Do.  26,  263  (1908);  16  pp. 
GUYE,  C.  E.  and  MOREIN,  A.     The  Interior  Friction  of  Quartz  Filaments 

at  High  Temperatures.     Arch.  Sci.  Phys.  Nat.  62,  351  (1920);  21  pp. 
GUYE,  C.  E.  and  SCHAPPER,  H.     Sur  le  frottement  int^rieur  des  m6taux 

aux   basses   temperatures.     Compt,    rend.    150,    963    (1910);    2   pp.; 

Arch.  sci.  phys.  nat.  30,  133  (1909);  19  pp. 
GUYE,  C.  and  VASSILEFF,  S.     (1)  Internal  Friction  of  Glasses  as  a  Function 


378  INDEX 

of  the  Temperature.     Arch.  sci.  phys.  nat.  37,  214  (1914);  11  pp.; 

(2)  Do.     Do   37,  301  (1914);  27  pp. 
GUTE,  P.  A.     Molecular  Complexity  in  Liquid  State.     J.  chim.  phys.  9, 

504  (1911);  4pp. 
GUYE,  P.  &  FRIDERICHS,  L.     Sur  la  mesure  des  coefficients  de  viscositc. 

Bull.  soc.  chim.  (3)  19,  164  (1898);  5  pp. 
GUZMAN,  J.  DE.     Relaci6n  entre  la  fluidez  y  el  calor  de  fusi6n.     Anales  soc. 

esp.  fis.  y  quim.     Pt.  I.  11,  353  (1913);  9  pp. 

DE  HAAS.  130,  Measurements  on  the  Influence  of  Temperature  on  the 
Viscosity  of  Methyl  Chloride  in  Absolute  Measure  between  the  Boiling- 
Point  and  the  Critical  State.  Diss.  Leiden  (1894);  Comm.  Leiden  #12. 

HABER.  (Viscosity  method  of  measuring  high  vacua.)  Z.  Elektrochem. 
20,  296  (1914). 

HACHETTE.  Sur  1'ecoulement  des  fluides.  Ann.  chim.  phys.  (2)  5,  52 
(1817);  7  pp.  Cp.  Phil.  Trans.  85  (1795). 

HADAMARD,  M.  29,  (1)  Sur  les  glissements  dans  les  fluides.  Compt.  rend. 
136,  299  (1903);  3  pp.;  (2)  Do.  Rectification  a  une  note  pr6ce"dente. 
Compt.  rend.  136,  545  (1903);  1  p. 

HAFFNER,  G.  tlber  die  innere  Reibung  von  alkoholischen  Salzlosungen. 
Diss.  Erlangen  (1903);  Physik.  Z.  2,  739  (1901). 

HAGEN,  G.  2,  18,  36,  (1)  tlber  die  Bewegung  des  Wassers  in  engen  cylin- 
drischen  Rohren.  Pogg.  Ann.  46,  423  (1839);  20  pp.;  (2)  Uber  den  Ein- 
fluss  der  Temperatur  auf  die  Bewegung  des  Wassers  in  Rohren.  Abh. 
d.  Berl.  Akad.  (1854);  81  pp. 

HAGENBACH,  E.  2,  13,  17,  18,  (1)  tlber  die  Bestimmung  der  Zahigkeit 
einer  Flussigkeit  durch  den  Ausfluss  aus  Rohren.  Pogg.  Ann.  109, 
385  (1860);  42  pp.;  (2)  Definition  de  la  viscosite  d'un  liquide.  Compt. 
rendu  des  travaux  de  la  societe  Helvetique  des  Sciences  naturelles  a 
Zermatt  (1895);  Archives  de  Geneve  (3)  34,  377  (1895). 

HALLOCK,  W.  The  Flow  of  Solids  or  the  Behavior  of  Solids  under  High 
Pressure.  Bull,  of  the  U.  S.  Geol.  Survey  56,  67  (1890);  8  pp.  Cp. 
Annual  Report  of  the  Board  of  Regents  of  the  Smithsonian  Inst. 
237  (1891). 

HANDOWSKY.  Biochem.  Z.  25,  510  (1910);  Koll.-Z.  7,  183  (1910);  Do. 
7,  267  (1910). 

HANNAY,  J.  B.  On  the  Microrheometer.  Phil.  Trans.  170,  275  (1879); 
(abstract)  Proc.  Roy.  Soc.  London  28,  279  (1879);  2  pp.  Cp.  Barnett. 

HAPPEL,  H.  Die  Edelgase.  Innere  Reibung,  Warmeleitung.  und  Diffusion. 
Phys.  Z.  10,  484  (1909);  5  pp. 

HARDY,  W.  B.  95,  213,  (1)  Colloidal  Solution.  The  Globulina.  J.  of 
Physiology  33,  251  (1905).  Proc.  Roy.  Soc.  B79,  413  (1907);  (2)  Prob- 
lems of  Lubrication.  Proc.  Roy.  Inst.  Gt.  Britian,  Feb.  27  (1920); 
8  pp. 

HARKINS,  W.  D.  257,  An  Apparent  High  Pressure  Due  to  Adsorpton. 
Proc.  Nat.  Acad.  Sci.  6,  49  (1920);  7  pp. 


INDEX  379 

HARD.  286,  (1)  Sur  1'ecoulement  du  sang  par  des  tubas  de  petit  calibre. 
Compt.  rend.  83,  696  (1876);  3  pp.;  (2)  Essai  sur  lu  transpirabilite  du 
sang.  Gaz.  Hebdom.  April  11  (1873);  (3)  Transpirabilite"  du  sang. 
Gaz.  Hebdom.  July  7  (1876). 

HARRISON,  W.  J.  (1)  The  Hydrodynamical  Theory  of  Lubrication  with 
Special  Reference  to  Air  as  a  Lubricant.  Trans.  Camb.  Phil.  Soc.  22, 39 
(1913);  15  pp.;  (2)  The  Motion  of  a  Viscous  Liquid  Due  to  Uniform 
&  Periodic  Motion  Maintained  over  a  Segment  of  an  Infinite  Plane 
Boundary.  Proc.  Roy.  Soc.  London  (7)  88,  13  (1913);  11  pp.;  (3) 
J.  Soc.  Dyers  and  Colourists  27,  April  (1911). 

HARTLEY,  H.,  THOMAS,  N.  and  APPLEBEY,  M.  196,  Some  Physico-Chemical 
Properties  of  Mixtures  of  Pyridine  and  Water.  Trans.  Chem.  Soc. 
93,  538  (1908). 

HASSELBLATE,  M.  Uber  die  lineare  Kristallisationsgeschwindigkeit  isom- 
orpher  Mischungen.  Z.  physik.  Chem.  83,  1  (1913);  39  pp. 

HATSCHEK,  E.  206,  (1)  The  Viscosity  of  Dispersed  Systems.  Z.  Chem. 
Ind.  Kolloide  7,  301  (1910);  4  pp.;  (2)  Kolloid.-Z.  7,  301  (1910);  (3) 
Do.  7,  81  (1910);  (4)  The  Viscosity  of  Disperiods.  Z.  Chem.  Ind. 
Kolloid  8,  34  (1911);  5  pp.  Cp.  C.  A.  6,  2458;  (5)  Kolloid.-Z.  11,  158 
(1912);  (6)  Die  Allegemeine  Theorie  der  Viskositat  zweiphasiger 
Systeme.  Z.  Chem.  Ind.  Kolloid.  12,  238  (1913);  8  pp.  Cp.  Trans. 
Faraday  Soc.  (1913);  The  General  Theory  of  Two-Phase  Systems, 
14  pp.;  (7)  The  Viscosity  of  Emulsoid  Sols,  and  Its  Dependence  on  the 
Rate  of  Flow.  Kolloid-Z.  13, 88  (1913) ;  8  pp. ;  (8)  Viscosity  of  Colloidal 
Solutions.  Proc.  Phys.  Soc.  London  28,  250  (1916).  Cp.  Humphrey; 
(9)  Viscosity  of  Colloids.  First  Report  on  Colloid  Chemistry,  Brit. 
Assoc.  for  the  Adv.  of  Sci.  (1917);  4  pp.;  H.  M.  Stationery  Office, 
Imperial  House,  Kingsway,  London,  W.  C.;  (10)  The  Composition 
of  the  Disperse  Phase  of  Emulsoid  Sols.  Kolloid-Z.  11,  284  (1912); 
(11)  The  Viscosity  and  Hydration  of  Colloidal  Solutions.  Biochem. 
J.  10,  325  (1916). 

HAUSER,  L.  140,  Uber  den  Einfluss  des  Druckes  auf  die  Viscositat  des 
Wassers.  Diss.  Tubingen  (1900).  Ann.  Physik.  (4)  6,  597  (1901). 

HECHLER,  W.  Fluiditat  und  Leitfahigkeit  einiger  konzentriter  wasseriger 
Salzlosungen  unten  0°.  Ann.  Physik.  (4)  16,  157  (1904);  16  pp. 

DE  KEEN,  P.  6,  131,  (1)  De  la  fluidite  des  liquides.  Bull,  de  1'Ac.  Roy. 
Belg.  (2)  45,  798  (1878)  19pp.;  (2)  Determination  d'une  relation  empiri- 
que  entre  le  coefficient  de  frottement  inte'rieur  des  liquides  et  les  varia- 
tions que  celui-ci  eprouve  avec  la  temperature.  Bull,  de  1'Ac.  Roy. 
Belg.  (3)  7,  248  (1884);  5  pp  ;  (3)  Determination  des  variations  que  le 
coefficient  de  frottement  interieur  des  liquides  e"prouve  avec  la  tempera- 
ture. Considerations  the"oriques  que  d£coulent  de  1'observation  de  ces 
grandeurs.  Bull  de  1'Ac.  Roy.  Belg.  10,  251  (1885);  15  pp.;  (4)  Do. 
Bull,  de  1'Ac.  Roy.  Belg.  (3)  11,  29  (1886);  16  pp.;  (5)  Determination 
des  variations  que  le  frottement  inte'rieur  de  1'air  pris  sous  diverses 
pressions  eprouve  avec  la  temperature.  Bull,  de  1'Ac.  Roy.  Belg. 


380  INDEX 

(3)  16,  195  (1888)  21  pp.;  Phil.  Mag.  (5)  28,  220  (1889);  (6)  Theorie 
des  liquides  (1888);  2  parts;  (7)  Recherches  sur  la  physique  comparee 
et  la  theorie  des  liquides.  Gauthier  Villars  (1888). 

HEFELMANN,  R.  (Viscosity  of  gum  arabic.)  Z.  offentl.  Chem.  7,  195 
(1901);  3  pp. 

HEFFT,  O.     Diss.  Heidelburg  (1900). 

HEFFTER,  A.  Uber  die  Ernahrung  des  arbeitenden  Froschherzens.  Arch, 
fur.  exp.  Path.  Pharm.  29,  41  (1892). 

HELE-SHAW,  H.  (1)  Flow  of  liquids  in  Thin  Films  Rep.  Brit.  Assoc.  for 
Adv.  of  Sci.  Contribution  a  l'e"tude  th6orique  et  exp6rimentale  des 
veines  liquides  deiorm^es  par  des  obstacles  et  a  la  determination  des 
lignes  d'induction  d'un  champ  magnelique.  Compt.  rend.  132,  1306 
(1901);  6  pp.;  (2)  Proc.  Inst.  of  Noval  Architects  (1898  and  1899). 

HELMHOLTZ,  H.  2, 14,  29,  (1)  Uber  Reibung  und  Warmeleitung  verdunnter 
Case.  Monatsber.  der  Konigl.  Akad.  der  Wissens.  zu  Berlin,  Feb.  25 
(1875);  14  pp.;  Wissens.  Abh.  Physik.-Tech.  Reichsanst  1,  158  (1896) 
also  collected  works,  p.  223;  (2)  Uber  den  Einfluss  der  Reibung  in  der 
Luft  auf  die  Schallbewegung.  Verh.  des  naturhistorisch.  medizinischen 
Vereins  zu  Heidelberg  17,  257  (1863);  4  pp. 

HELMHOLTZ,  H.  und  PIOTROWSKI,  G.  6,  30,  Uber  Reibung  tropfbarer 
Fliissigkeiten.  Wien.  Sitzungsber.  (2A)  40,  607  (1868);  52  pp.  Cp. 
Ladenburg. 

DE  HEMPTINNE,  A.  Transforming  Animal  and  Vegetable  Oils  into  Viscous 
Products.  Brit.  pat.  15,  748,  July  6  (1909). 

HENDERSON,  L.  J.,  FENN,  W.  O.  and  COHN,  E.  J.  Influence  of  Electrolytes 
upon  the  Viscosity  of  Dough.  J.  Gen.  Physiology  1,  387  (1919);  10 
pp. 

HENNESSEY,  H.  On  the  Maximum  Discharge  through  a  Pipe  of  Circular 
Section  when  the  Effective  Head  is  Due  only  to  the  Pipe's  Inclination. 
Proc.  Roy.  Soc.  London  46,  145  (1888);  3  pp. 

HENRI,  V.  The  Determination  of  Size  of  Colloidal  Particles.  Trans. 
Faraday  Soc.  (1913);  7  pp. 

HENRI,  V.,  LALOU,  MAYER,  A.  and  STODEL.  Sur  les  ph6nomenes  qui 
precedent  la  precipitation  des  colloides  par  les  electrolites  et  sur  les 
moyens  de  les  mettre  en  evidence.  Compt.  rend.  Soc.  Biol.  65,  1668 
(1903). 

HENRY,  ET  CALUGAREANU.  213,  Diffusion  des  matieres  colorantes  dans  la 
gelatine  et  dans  1'eau.  Compt.  rend.  Soc.  Biol.  579  (1901). 

HENRY,  V.  and  MAYER,  A.  Variations  des  albuminoides  du  plasma  sanguin 
au  cours  du  lavage  du  sang.  Compt.  rend.  Soc.  Biol.  64,  824  (1902). 

HERAEUS,  W.  C.    Zeitschr.  f.  angewandte  chem.  18,  49  (1905). 

HERSCHEL,  W.  H.  324,  329,  (1)  Determination  of  Absolute  Viscosity  by  the 
Saybolt  Universal  and  the  Engler  Viscometers.  Proc.  Am.  Soc.  Testing 
Materials  II,  17,  551  (1917);  20  pp.;  (2)  Standardization  of  the  Saybolt 
Universal  Viscometer.  U.  S.  Bur.  of  Standards  Tech.  Paper  112 
(1918);  25  pp.;  (3)  A  Viscometer  for  Gasoline.  Proc.  Am.  Soc.  Testing 


INDEX  381 

Materials  II,   19,  676   (1919);   11  pp.;  U.  S.  Bur.  of  Standards  Tech. 

Paper  125  (1919);  18  pp.;  (4)  Saybolt  Viscosity  of  Oil  Blends.     Chem. 

Met.  Eng.  22,  1109  (1920);  3  pp.;  U.  S.  Bur.  of  Standards  Tech.  Paper 

164  (1920);  (5)  The  MacMichael  Tortional  Viscometer.     J.  Ind.  Eng. 

Chem.  12,  282  (1920);  6  pp. 

HERSCHEL,  W.  H.  and  BERGQUIST,  C.     The  Consistency  of  Dextrin  Pastes. 
HERSEY,  M.  D.     The  Theory  of  the  Torsion  and  Rolling-Ball  Viscometers 

and  Their  Use  in  Measuring  the  Effect  of  Pressure  on  Viscosity.     J. 

Wash.  Acad.  Sci.  6,  525  (1916);  6  pp. 
HERZ,    W.     (1)    Internal    Friction   of   Aqueous    Solutions   of    Potassium 

Halide  Solutions.     Z.  anorg.  Chem.  86,  338  (1914);  2  pp.;  (2)  Internal 

Friction  of  Chlorinated  Hydrocarbons.     Z.  Elektrochem.  23,  24  (1917); 

J  Chem.  Soc.  II,  112, 194;  (3)  Internal  Friction  of  Aqueous  Salt  Solutions. 

Z.  anorg.  allgem.  Chem.  99, 132  (1917);  (4)  Fluidity  and  Specific  Volume 

of  Aqueous  Solutions.     Do.  102,  173  (1918);  3  pp.;  J.  Chem.  Soc.  II, 

114,  155  (1918);  (5)  Do.,  do.  104,  47  (1918);  6pp. 

HERZ,  W.  and  RATHMANN,  W.     The  Inner  Friction  of  Chlorinated  Ali- 
phatic   Hydrocarbons    and    their    Mixtures.     Z.     Elektrochem.     19, 

589  (1913);  1  p. 
HERZOG,  R.  O.     Viscosity  of  Colloidal  Sols.     Z.  Chem.  Ind.  Kolloide  8, 

210  (1911);  2  pp. 

HESS,  A.     Apparent  Viscosity  of  Dielectrics.     Eclair.  Elect.  7,  450  (1896). 
HESS,    H.     239,    (1)    Elasticitat    und    innere    Reibung    des    Eises.     Ann. 

Physik.  (4)  8,  405  (1902);  27  pp.;  (2)  The  Plasticity  of  Ice.     Ann. 

Physik.  36,  449  (1911);  44  pp. 
HESS,  W.  R.     (1)   Viskositat  des  Blutes  und  Herzarbeit.     Diss.  Zurich 

(1906);    12   pp.;    Vierteljahrschrift   der  Naturforsch.  Ges.  Zurich  51, 

(1906);  (2)  Ein  Neuer  Apparat  zur  Bestimmung  der  Viskositat  des 

Blutes.     Munch,  med.  Wochschr.   #32   (1907);   (3)  Die  Bestimmung 

der  Viskositat  des  Blutes.     Miinch.  med.  Wochschr.  #45  (1907);  (4) 

Die    Viskositat    des    Blutes    bei    gesunden    Menschen.     Deut.    Arch. 

klin.  Med.  94  (1908);  (5)  Theory  of  Viscosity  of  Heterogeneous  Systems. 

Kolloid-Z.  27,  1  (1920);  12  pp.;  (6)  Viscosity  of  Gel-Forming  Solutions. 

Do.  27,  154  (1920);  10pp. 
VON  HEVESY,  G.     Mobility  of  Ions  which  are  the  same  as  those  of  the 

Solvent.     Z.  Elektrochem.  27,  21  (1921);  4  pp. 
HEYDWEILLER,  A.     7,  130,  142,  235,  (1)  Der  Temperatureinfluss  auf  die 

innere  Reibung  von  Benzol  und  A"  thylather  oberhalb  ihres  Siedepunktes. 

Wied.  Ann.  55,  561   (1895);   15  pp.;   (2)  Die  innere  Reibung  einiger 

Fliissigkeiten  oberhalb  ihres  Siedepunktes.     Wied.  Ann.  59,  193  (1896); 

20  pp. ;  (3)  Zur  Bestimmung  der  inneren  Reibung  fester  Korper.     Wied. 

Ann.  63, 56  (1897);  4  pp. 
HEYMANN,  T.     Diss.  Zurich  (1901). 
HIGGINS,   W.   F.     324,   (1)   Methods  and  Apparatus  Used  in  Petroleum 

Testing.     II.  Viscosity.     J.  Soc.  Chem.  Ind.  32,  568  (1913);  5  pp.; 

Also  Petroleum  World  June   (1913);  National  Physical  Laboratory, 

Collected  Researches  11,  1  (1914);  16  pp. 


382  INDEX 

HIGGINS,  E.  F.  and  PITMAN,  E.  C.  Measurement  of  the  Viscosity  of 
Pyroxylin  Solutions.  J.  Ind.  Eng.  Chem.  12,  587  (1920);  5  pp. 

HILDITCH,  T.  &  DUNSTAN,  A.  E.  Ill,  (1)  Correlation  of  Viscosity  with 
other  Constitutive  Properties.  Proc.  Chem.  Soc.  26,  341  (1910);  1  p.; 
(2)  Die  Beziehung  der  Viscositat  zu  anderen  physikalischen  Eigen- 
schaften  I,  Athan-und  Athinverbindungen.  Z.  Elektrochem.  17,  929 
(1911);  5  pp.;  (3)  II.  Einfluss  der  Anlagerung  ungesattigter  Gruppen. 
Do.  18,  185  (1912);  5  pp.;  (4)  III.  Der  Einfluss  von  raumlichbenach- 
barten  ungesattigten  Gruppen.  Do.  18,  881  (1912);  5  pp.;  (5)  The 
Relation  of  Viscosity  to  other  Physical  Properties.  Z.  Elektrochem. 
17,  929  (1912);  5  pp.  Cp.  Dunstan. 

HIMSTEDT,  F.  tJber  das  Zusammenwirken  von  Zug  und  Torsion  bei 
Metalldrahten.  Wied.  Ann.  17,  701  (1882);  12  pp. 

HIRN,  G.  A.  tfber  die  hauptsachl.chsten  Erscheinungen  der  mittelbaren 
Reibung.  Dingl.  polytech.  J.  136,  405  (1885);  10  pp. 

HIRSCH,  C.  &  BECK,  C.  (1)  Eine  Methode  zur  Bestimmung  des  inneren 
Reibungswiderstandes  des  lebenden  Blutes  beim  Menschen.  Munch. 
Med.  Wochenschr.  47,  1685  (1900);  2  pp.;  (2)  Studien  zur  Lehre  von  der 
Viscositat  (inneren  Reibung)  des  lebenden,  menschlichen  Blutes. 
Deutsch.  Arch.  f.  klin.  Med.  69,  503  (1901);  18  pp.;  (3)  Do.,  do. 
72,  560  (1902).  Cp.  Beck. 

HOFFMANN,  P.  tTber  die  Stromung  der  Luft  durch  Rohren  von  beliebiger 
Lange.  Diss.  Breslau  (1883);  Wied.  Ann.  21,  470  (1884);  24  pp. 

HOFSASS,  M.  (1)  Apparatus  for  Determining  the  Density  of  a  Gas  and  a 
Viscometer  for  Gases.  J.  Gasbel.  66,  841  (1913);  2  pp.;  (2)  The  Vis- 
cosity of  Gases.  Do.  62,  776  (1919);  2  pp. 

HOGG,  J.  242,  (1)  Viscosity  of  Air.  Proc.  Am.  Acad.  40,  611  (1905);  16 
pp.;  (2)  Contrib.  Jeff.  Phys.  Lab.  2,  611  (1904);  (3)  Friction  in  Gases  at 
low  Pressure.  Proc.  Am.  Acad.  42,  113  (1906);  33  pp.;  (4)  Friction  in 
Gases  at  low  Pressure.  Proc.  Am.  Acad.  46,  1  (1910);  16  pp.;  Phil. 
Mag.  (6)  19,  376  (1910);  14  pp. 

HOLDE,  D.  (1)  Bericht  uber  vergleichende  Schmieroluntersuchungen, 
ausgefiihrt  in  den  Jahren  1889-1894.  Mitt.  Konigl.  techn.  Versuch- 
anstatt.  Erganzungsheft  I,  Springer  (1895);  78  pp.;  (2)  Ober  kolloide 
Losungen  von  Kalkseifen  in  schweren  Mineralolen.  Koll.  Zeitschr. 
3,  270  (1908);  5  pp.  Cp.  Konigl  Mat.  Prtif.  Amt.  Jahresber.  (1905) 
Mitteil  (1906);  (3)  Untersuchung  der  Mineralolen  und  Fette.  3d. 
Ed.  (1909);  Springer,  Berlin;  (4)  The  Determination  of  the  Viscosity 
of  Liquid  Lubricants.  Orig.  Com.  8th  Intern.  Congr.  Appl.  Chem. 
(appendix)  26,  677  (1913);  4  pp. 

HOLKER,  J.  .(1)  The  Viscometer  as  a  Means  for  Determining  Specific 
Gravity.  J.  Path.  Bact.  23,  185  (1920);  3  pp.;  (2)  A  Method  for 
Determining  Several  Viscosities  Simultaneously.  Do.  23,  177  (1920); 
8pp. 

HOLLAND,  R.  Uber  die  Anderung  der  electrischen  Leitfahigkeit  einer 
Losung  durch  Zusatz  von  kleinen  Mengen  eines  Nichtleiters.  Wied. 
Ann.  60,  261  (1893);  32  pp. 


INDEX  383 

HOLMAN,  S.     246,  247,  (1)  On  the  Effect  of  Temperature  on  the  Viscosity 

of  the  Air.     Prdc.  Am.  Acad.  12,  41  (1876);  10  pp.;  (2)  A  New  Method 

of  Studying  the  Relation  between  the  Viscosity  and  Temperature  of 

Gases.     Phil.  Mag.  (5)  3, 81  (1877) ;  6  pp. ;  (3)  On  the  Effect  of  Tempera- 
ture on  the  Viscosity  of  the  Air.     Proc.   Am.  Acad.   21,  1   (1885); 

(4)  On  the  Effect  of  Temperature  on  the  Viscosity  of  Air  and  Carbon 

Dioxide.     Phil.  Mag.  (5)  21,  199  (1886);  24  pp. 
HOLMGREN,  I.     The  Influence  of  White  Blood  Corpuscles  upon  the  Viscosity 

of  the  Blood.     Deut.  Med.  Wochschr.  39,  217  (1913);  2  pp. 
HONDA,  K.  and  KONNO,  S.     On  the  Determination  of  the  Coefficient  of 

Normal    Viscosity    of    Metals.     Phil.     Mag.    42    (6)     115     (1921); 

8pp. 

HORIBA.     J.  Tok.  Chem.  Soc.  31,  922  (191  ). 
HORTON,  F.     237,  The  Effects  of  Changes  of  Temperature  on  the  Modulus 

of  Torsional  Rigidity  of  Metal  Wires.     Phil.  Trans.  London  (A)  204, 

1  (1904);  55pp. 
HOSKING,  R.     6,  (1)  Viscosity  of  Solutions.     Phil.  Mag.  (5)  49,  274  (1900); 

13  pp.     Cp.  Lyle  &  Hosking;   (2)  The  Electrical  Conductivity  and 

Fluidity  of  Solutions  of  Lithium  Chloride.     Phil.   Mag.    (6)   7,  469 

(1904) ;  29  pp. ;  (3)  The  Viscosity  of  Water.     J.  Proc.  Roy.  Soc.  N.  S.  W. 

42,  34  (1909);  23  pp.     Do.,  do.  43,  34  (1910);  5  pp. 
HOUBA.     Over    de    strooming    van    vloeistoffen    door   buizen.     Nijmegen 

(1883).     Cp.  Wied.  Ann.  21,  493  (1884). 
HOUDAILLE,  F.     Mesure  du  coefficient  de  diffusion  de  la  vapeur  d'eau  dans 

1'atmosphere  et  du  coefficient  de  frottement  de  la  vapor  d'eau.     These 

Paris  (1896);  Fortsch.  Physik.  (I)  52,  442  (1896);  1  p. 
HOWARD,  W.  B.     Penetration  Needle  Apparatus  for  Testing  the  Viscosity 

of  Asphalt.     U.  S.  Pat.  1,225,438,  May  8  (1917). 
HOWEL  and  COOKE.     Action  of  the  Inorganic  Salts,  of  Serum,  Milk,  Gastric 

Juice,  etc.,  upon  the  Isolated  Heart,  etc.     J.  of  Physiol.  14,  198  (1893). 
HUBBARD,  P.  &  REEVE,  C.     Methods  for  the  Examination  of  Bituminous 

Road  Materials.     U.  S.  Dept,  Agric.  Bull.  No.  314.     Cp.  also  Eng. 

Contr.  54,  16  (1920);  4  p. 
HUBENER,  T.     17.8,  179,  Untersuchungen  iiber  die  Transpiration  von  Salz- 

losungen.     Pogg.  Ann.  150,  248  (1873);  12  pp. 
HUBNER,  W.     286,  Die  Viscositat  des  Blutes.     Bemerkungen  zu  der  gleich- 

namigen  Arbeit  von  C.  Beck.  u.  C.  Hirsch.     Arch.  f.  exp.  Path.  u. 

Pharm.  54,  149  (1905). 
HURTHLE,  K.     (1)  Uber  den  Widerstand  der  Blutbahn.     Deutsch.  med. 

Wochschr.  23,  #51,  809  (1897);  3  pp.;  (2)  Do.     Arch.  ges.  Physiol. 

(Pfluger's)  82,  415  (1908);  3  pp. 
HUMPHREY,  E.  and  HATSCHEK,  E.     The  Viscosity  of  Suspensions  of  Rigid 

Particles  at  Different  Rates  of  Shear.     Proc.  Phys.  Soc.  London  28, 

274  (1916). 

HURST.     Lubricating  Oils,  Fats,  and  Waxes.     3d  Ed.  (1911). 
HUTCHINSON,    J.     Viscometer   for   Use   with    Coal-Tar,    etc.     Brit.    Pat. 

22,042,  Oct.  6  (1911). 


384  INDEX 

HYDE,  J.  H.  141,  (1)  The  Construction  of  an  Apparatus  for  the  Determi- 
nation of  the  Absolute  Viscosities  of  Liquids  at  High  Pressures  and  the 
Results  Obtained  with  it  for  certain  Lubricating  Oils.  Dept.  Sci. 
Ind.  Research  Advisory  Council.  Report  of  the  Lubricants  and 
Lubrication  Inquiry  Comm.  (1920);  5  pp.;  (2)  The  Absolute  Viscosity 
of  Liquids  at  High  Pressure.  Do.,  2  pp.;  (3)  The  Determination  of  the 
Compressibility  of  Lubricating  Oils  under  High  Pressure.  Do.,  5  pp.; 
(4)  Improvement  of  the  Lubricating  Properties  of  Mineral  Oils.  Engi- 
neering 111,  708  (1920);  1  p.;  (5)  The  Viscosities  and  Compressibilities 
of  Liquids  at  High  Pressures.  Proc.  Roy.  Soc.  London  (A)97,  240 
(1920);  10pp. 

HYDEN,  W.  L.     291,  et  seq.,  Thesis  Lafayette  College  (1921). 

IBBETSON,  W.     An  Elementary  Treatise  on  the  Mathematical  Theory  of 

Perfectly   Elastic   Solids,   with   a   Short  Account  of   Viscous  Fluids. 

Macmillan  Company,  London  and  New  York  (1887);  515  pp. 
IOKIBE,  K.  and  SAKAI,  S.     The  Effect  of  Temperature  on  the  Modulus  of 

Rigidity  and  on  the  Viscosity  of  Solid  Metals.     Phil.   Mag.  42   (6) 

397  (1921);  22  pp. 
ISHIMOTO,    M.     Investigation   of    Metals   with    regard    to    their    Internal 

Friction.     Phys.  Soc.  Japan  (3)  1,  267  (1919);  10  pp. 
ISHIZAKA,  N.     Uber  die  Beziehung  zwischen  Kolloidfallung  und  Adsorption 

und    tiber   die .  Fallungsgeschwindigkeit.     Z.    physik.    Chem.    83,    97 

(1913);  31  pp. 
ISRAEL,  H.     Theorie  der  Ausfliisszeiten  einer  Fliissigkeit.     Diss.  Rostock 

(1905);  65  pp. 
IZAR,  G.     Lowering  of  Viscosity  by  Gelatine  Antiserum.     Z.  Immunitat 

7,  199  (1909);  5  pp. 

JABLCZYNSKI,  K.  The  Velocity  of  the  Formation  of  Precipitates.  Z. 
physik.  Chem.  82,  115  (1913);  6  pp. 

JACKSON,  H.  Glass  and  Some  of  its  Properties.  J.  Roy.  Art  68,  134 
(192Q);  13pp. 

JACOBSON,  H.  2,  14,  17,  32,  (1)  Beitrage  zur  Haemodynamik.  Arch.  f. 
Anat.  und  Phys.  80  (1860);  33  pp.;  (2)  Zur  Einleitung  in  die  Haemo- 
dynamik. Arch.  f.  Anat.  und  Phys.  304  (1861);  25  pp.  Cp.  Ber.  d. 
Naturf.  Vers.  in  Konigsberg  (1862)  (1867). 

JAGER,  G.  131,  (1)  tlber  die  kinetische  Theorie  der  inneren  Reibung  der 
Flussigkeiten.  Wien.  Sitzungsber.  (2A)  102,  253  (1893);  12  pp.; 
(2)  t)ber  die  innere  Reibung  der  Losungen.  Wien.  Sitzungsber.  251 
(1894);  15  pp.;  (3)  Uber  den  Einfluss  des  Molecularvolumens  auf  die 
innere  Reibung  der  Case.  Wien.  Sitzungsber.  (2A)  108,  447  (1899); 
9  pp.;  (4)  Do.  Wien.  Sitzungsber.  (2A)  109,  74  (1900);  7  pp.  Cp. 
Wien.  Anzeiger  Kaiserl.  Akad.  Wissens.  math.-naturw.  Kl.  11  (1900); 
1  p.;  (5)  Der  innere  Druck,  die  innere  Reibung  die  Grosse  der  Molekeln 
und  deren  mittlere  Weglange  bei  Flussigkeiten.  Wien.  Sitzungsber. 
(2A)  111,  697  (1902);  10  pp.;  (6)  Kinetische  Theorie  der  Gasen.  Hand- 


INDEX  385 

buch  der  Physik.  Winklemann.  2  ed.  3,  734  (1906);  13  pp.;  (7)  The 
Kinetic  Theory  of  the  Internal  Friction  of  Gases.  Sitz.  Akad.  Wiss. 
Wien.  (IIA)  127,  849  (1918);  22  pp. 

JAPPELI,  G.  Contribute  allo  studio  dell'influenza  della  aumentata  viscosita 
del  sangue  sulla  meccanica  cardio-vascolare.  Arch,  di  Fisiologia  4, 
101  (1907).  Cp.  Botazzi. 

JEANS.  The  Dynamical  Theory  of  Gases.  Camb.  Univ.  Press  (1904); 
347  pp. 

JEAUCARD  &  SATIE.  Tension  superficielle  et  viscosite  de  quelques  huiles 
essentielles.  Bull.  Soc.  chim.  (3)  25,  519  (1901);  5  pp. 

JEVONS,  W.  S.  On  the  Movement  of  Microscopic  Particles  Suspended  in 
Liquids.  Quarterly  J.  of  Science,  London  (1878);  22  pp. 

JOB.  Nouvelle  methode  experimentale  pour  1'etude  de  la  transpira- 
tion des  gaz.  Soc.  franc,  d.  phys.  157,  2  (1901).  Cp.  Fortsch.  Physik. 
67,  280  (1901). 

JOHNSON  &  BLAKE.  On  Kaolinite  and  Pholerite.  Amer.  J.  Sci.  (2)  93, 
351  (1867);  llpp. 

JOHNSTON,  J.  195,  196,  (1)  The  Change  of  the  Equivalent  Conductance  of 
Ions  with  the  Temperature.  J.  Am.  Chem.  Soc.  31, 1010  (1909);  11  pp.; 
(2)  A  Correlation  of  the  Elastic  Behavior  of  Metals  with  Certain  of 
their  Physical  Constants.  J.  Am.  Chem.  Soc.  34,  788  (1912);  15  pp. 

JOHNSTON,  J.  &  ADAMS,  L.  H.  On  the  Effect  of  High  Pressures  on  the 
Physical  and  Chemical  Behavior  of  Solids.  (Effect  of  Pressure  on 
Viscosity,  p.  229.)  Am.  J.  of  Sci.  35  (4)  205  (1913);  48  pp. 

JONES,  G.  C.     Ann.  Reports  on  the  Progress  of  Chem.  9,  195  (1913);  1  p. 

JONES,  H.  C.  &  COLLABORATORS.  The  Freezing-Point  Lowering,  Conduc- 
tivity  and  Viscosity  of  Solutions  of  certain  Electrolytes  in  Water. 
Methyl  Alcohol,  Ethyl  Alcohol,  Acetone  and  Glycerol,  Carnegie  Publ. 
180.  Cp.  Davis. 

JONES,  H.  C.  &  BINGHAM,  E.  C.  The  Conductivity  and  Viscosity  of 
Solutions  of  Certain  Salts  in  Mixtures  of  Acetone  with  Methyl  Alcohol, 
with  Ethyl  Alcohol,  and  Water.  Am.  Chem.  J.  34,  481  (1905); 
Cp.  Bingham. 

JONES,  H.  &  BINGHAM,  E.  &  McMASTER,  L.  Cp.  Jones  and  Bingham;  and 
Jones  and  McMaster.  Z.  physik.  Chem.  67,  193  (1906);  115  pp. 

JONES,  H.  &  CARROLL,  C.  A  Study  of  the  Conductivities  of  Certain  Elec- 
trolytes in  Water,  Methyl  and  Ethyl  Alcohola,  and  Mixtures  of  these 
Solvents — Relations  between  Conductivity  and  Viscosity.  Am.  Chem. 
J.  32,  521  (1904);  63  pp.;  Z.  physik.  Chem.  57,  257  (1906);  63  pp. 

JONES,  H.,  LINDSAY,  C.,  CARROLL,  C.,  BASSETT,  H.,  BINGHAM,  E.,  ROUILLER, 
C.,  MCMASTER,  L.  &  VEAZEY,  W.  Conductivity  and  Viscosity  of 
Mixed  Solvents.  Carnegie  Institution  of  Washington,  Publ.  80 
(1907),  227  pp. 

JONES,  H.  &  MAHIN,  E.  Conductivity  and  Viscosity  of  Dilute  Solutions 
of  Certain  Salts  in  Water,  Methyl  Alcohol,  Ethyl  Alcohol,  Acetone, 
and  Binary  Mixtures  of  these  Solvents.  Am.  Chem.  J.  36,  325  (1906); 
85  pp.;  Conductivity  and  Viscosity  of  Dilute  Solutions  of  Lithium 


386  INDEX 

Nitrate  and  Cadmium   Iodide  in   Binary  and  Ternary    Mixtures  of 

Acetone  with  Methyl  Alcohol,  Ethyl  Alcohol,  and  Water.     Z.  Physik. 

Chem.  69,  389  (1909);  30  pp.     Cp.  Guy. 
JONES,  H.  &  VEAZEY,  W.     183,  Possible  Explanation  of  the  Increase  in 

Viscosity  when  Alcohols  are  Mixed  with  Water  and  of  the  Negative 

Viscosity  Coefficients  of  Certain  Salts  when  Dissolved  in  Water.     Am. 

Chem.  J.  37,  405  (1907);  5  pp. 
JONES,  O.  G.     6,  The  Viscosity  of  Liquids.     Phil.  Mag.  (5)  37,  451  (1894); 

12pp. 
JOHNS.     Studien  zur  Viscositat  des  menschlichen  Blutes  beim  Gesunden 

und  Kranken.  Med.  Klinik  #28  (1909). 

KANITZ,  L.  F.     Einige  Bermerkungen  iiber  Coulomb's  Verfahren  die  Coha- 

sion  der  Flussigkeiten  zu  bestimmen.     Pogg.  Ann.  70,  74  (1847);  4  pp. 

Cp.  Moritz. 
KAESS.     Untersuchungen    tiber    die    Viscositat    des    Blutes    bei    Morbus 

Bosedowi  (1913);  17pp. 
KAGAN,  G.     Zur  Technik  der  Viskositatsbestimmung.     Inaug.  Diss.  Bern 

(1911);  24pp. 
KAHLBAUM,    W.     Uber   die    Durchgangsgeschwindigkeit   verdiinnter  Luft 

durch   Glasrohren  verschiedenen    Durchmessers.     Verhandl.    d.    Ges. 

deutscher  Naturforscher,  Ntirnberg  1893,  56  (1894). 
KAHLBAUM,   G.   &  RABER,   S.     Die  Konstante  der  inneren  .  Reibung  des 

Ricinusols  und  das  Gesetz  ihrer  Abhang  von  der  Temperatur.     Nova 

Acta,  Abhand.  Kaiserl.  Leop.  Carol,  deutsch.  Akad.  Naturf.  84,  203 

(1905).     Cp.  Raber. 

KAHRS,  F.     Viscometer.     U.  S.  Pat.  1,062,159,  May  20  (1913). 
KALMUS,    H.     Electrical    Conductivity    and    Viscosity    of    Electrolytes. 

(1906);  54  pp.;  Electrical  Conductivity  and  Viscosity  of  Some  Fused 

Electrolytes.     Diss.  Zurich  (1906);  54  pp.     Cp.  Lorenz  and  Kalmus, 

and  Goodwin  and  Kalmus. 
KAMMERER.     Mitteilung   iiber  Forschungsarbeiten  auf  dem   Gebiete  des 

Ingenieurwesens.     Ver.     d.     Ingenieure.     Heft     132,     Berlin    Julius 

Springer  (1913). 
KANITZ,  A.     179,  tlber  die  innere  Reibung  von  Salzlosungen  und  ihren 

Gemischen.     Z.  physik.  Chem.  22,  336  (1897);  21  pp. 
KANN,  L.     tJber  die  innere  Reibung  des  Broms  und  dessen  Abhangigkeit 

von  der  Temperatur.     Wien.  Sitzungsber.  (2A)  106,  431  (1897);  5  pp. 
KAPFF.     Die  Reibung  von  Schmierolen  bei  hoheren  Warmegraden.     Kraft, 

und  Licht,  Dusseldorf  7,  126  (1901);  2  pp. 
KAPLAN,  V.     The  Laws  of  Flow  with  regard  to  Fluidity  and    Friction. 

Z.  Ver.  deut.  Ing.,  Sept.  28  (1912). 
KARIYA,  S.     Influence  of  Adrenaline  on  Viscosity  of  the  Blood  in   Acute 

Beriberi.     Mitt.  Med.  Ges.  Tokio,  26,  #15  Zentr.  Biochem.   Biophys. 

12,  372. 
VON  KARMIAN,  T.     The  Viscosity  of  Liquids  in  the  State  of  Turbulent  Flow. 

Physik.  Z.  12,  283  (1911);  2  pp. 


INDEX  387 

KARSS,   W.     Untersuchungen  liber  die  Viscositat  des  Blutes  bei   Morbus 

Basedowi.     Diss.  Heidelberg  (1912);  17  pp. 
KASSEL,    R.     Viskositat   binarer    Fliissigkeitgemischen.      Cp.    Drucker   & 

Kassel.     Diss.  Leipzig  (1910) ;  50  pp. 

KATZENELSOHN,  N.     Diss.  Berlin  (1867);  Wied.  Beibl.  12,  307  (1888). 
KAWALKI,   W.     (1)   Untersuchungen  iiber  die  Diffusionsfahigkeit  einiger 

Electrolyte  in  Alcohol.     Ein  Beitrag  zur  Lehre  von  der  Constitution 

der  Losungen.     Wied.  Ann.  52,  166  and  300  (1894);  25  pp.,  28  pp.; 

(2)  Die  Abhangigkeit  der  Diffusionsfahigkeit  von  der  Anfangsconcen- 

tration  bei  verdunnten  Losungen.     Wied.  Am.  69,  637    (1896);  15  pp. 
KAWAMURA,   S.     Measurements  of  Viscosity  Particularly  Fitted  for  the 

Study  of  the  Coagulation  Phenomena  of  Al(OH)s.     J.  Coll.  Science 

Imp.  Univ.  of  Tokyo,  Japan  26,  #8  (1908);  29  pp. 
LORD  KELVIN  (THOMSON,  SIR.  W.).     218,  238,  (1)  On  the  Elasticity  and 

Viscosity  of  Metals.     Phil.  Mag.  (4)  30,  63  (1865);  9  pp.;  Do.,  Proc. 

Roy.  Soc.  London  14,  289  (1865);  9  pp.;  (2)  Stability  of  Fluid  Motion. 

Rectilinear  Motion  of  a  Viscous  Fluid  between  two  Parallel  Planes.    Phil. 

Mag.  (5)  24,  188  (1887);  9  pp.;  (3)  Stability  of  Motion.     Broad  River 

flowing  down  on  Inclined  Plane  Bed.     Do.  (5)  24,  272  (1887);  6  pp.; 

(4)  On  the  Propagation  of  Laminar  Motion  through  a  Turbulently 
Moving  Inviscid  Liquid.     Phil.  Mag.  (5)  24,  342  (1887);  12  pp.;  (5) 
On  the  Stability  of  Steady  and  of  Periodic  Fluid  Motion.     Phil.  Mag. 

(5)  23,  459   (1887);  6  pp.;   (6)  On  the  Stability  of  Steady  and  of 
Periodic  Fluid  Motion.     Phil.  Mag.  (5)  23,  529  (1887);  11  pp. 

KENDALL,  J.  92,  104,  105,  (1)  The  Viscosity  of  Binary  Mixtures.  Medd. 
K.  Vetenskapsakad.  Nobelinst.  2,  #25,  1  (1913);  16pp.;  (2)  The  Exten- 
sion of  the  Dilution  Law  to  Concentrated  Solutions.  J.  Am.  Chem. 
Soc.  36,  1069  (1914);  20pp. 

KENDALL,  J.  and  BRAKELEY.  Compound  Formation  and  Viscosity  in 
Solutions  of  the  Types  Acid:  Ester,  Acid:  Ketone  and  Acid:  Acid. 
J.  Am.  Chem.  Soc.  43,  1826  (1921);  9  pp. 

KENDALL,  J.  and  MONROE,  K.  P.  (1)  Viscosity  of  Liquids.  Viscosity- 
Composition  Curves  for  Ideal  Liquid  Mixtures.  J.  Am.  Chem.  Soc. 
39,  1787  (1917);  15  pp.;  (2)  Ideal  Solutions  of  Solids  in  Liquids.  Do. 
39,  1802  (1917);  4  pp.;  (3)  Ideality  of  the  System:  Benzene-Benzyl 
Benzoate  and  the  Validity  of  the  Bingham  Fluidity  Formula.  Do. 
43, 115  (1921); 11  pp. 

KENDALL,  J.  and  WRIGHT,  A.  H.  168,  Ideal  Mixtures  of  the  Types  Ether- 
Ether  and  Ester-Ether.  J.  Am.  Chem.  Soc.  42,  1776  (1920);  9  pp. 

KEPPELER,  G.  &  SPANGENBERG,  A.  Increasing  the  Plasticity  and  Binding 
Power  of  Clay,  Kaolin,  etc.  U.  S.  Pat.  1,013,603. 

KERNOT,  G.  and  POMILIO,  U.  Cryoscopic  and  Viscometric  Behavior  of  Some 
Quinoline  Solutions.  Rend.  Acad.  Sci.  fis.  mat.  Napoli  17,  358 
(191-);  15pp. 

KHARICHKOV,  K.  V.  Uber  den  Einfluss  des  Wassers  auf  den  Entflammungs- 
punkt  und  Viskositat  von  Mineralschmierol  und  Naphtaruckstanden. 
Chem.  Ztg.  Rep.  376  (1907). 


388  INDEX 

KILLING,  C.  (1)  Eine  einfache  Methode  zur  Untersuchung  von  Butter 
auf  fremde  Fette.  Z.  angew.  Chem.  642  (1894);  3  pp.;  (2)  Do.  Chem. 
Ztg.  22,  78,  100  (1898);  (3)  Do.  Chem.  Rev.  Fettind.  9,  202  (1902). 

KING,  L.  V.  Turbulent  Flow  in  Pipes  and  Channels.  Phil.  Mag.  31,  322 
(1916). 

KINGSBURY,  A.  (1)  Trans.  Am.  Soc.  M.  E.  17,  116  (1895);  (2)  Experiments 
with  an  Air  Lubricated  Journal.  J.  Am.  Soc.  Nav.  Eng.  9,  #2  (1897); 
29pp. 

KINNISON,  C.  S.  A  Study  of  the  Atterberg  Plasticity  Method.  Trans. 
Am.  Ceram.  Soc.  16,  472  (1914);  13  pp.;  Tech.  Paper,  TJ.  S.  Bur.  of 
Standards  #46  (1915). 

KIRCHOF,  F.     Math.  Physik.  26  Vorelesung.  Mechanik  2  Aufl.  370  (1877). 

KIRCHOF,  F.  The  Influence  of  the  Solvent  on  the  Viscosity  of  India-Rubber 
Solutions.  Kolloid-Z.  15,  30  (1914). 

KIRKPATRICK,  F.  A.  and  ORANGE,  W.  B.  Tests  of  Clays  and  Limes  by  the 
Bureau  of  Standards  Plasticimeter.  J.  Am.  Ceram.  Soc.  1,  170  (1918); 
15pp. 

KOCH,  K.  R.  35,  External  Friction  in  Liquids.  Ann.  Phys.  35,  613  (1911) ; 
4pp. 

KOCH,  S.  (1)  t)ber  die  Abhangigkeit  der  Reibungskonstanten  des  Queck- 
silbers  von  der  Temperatur.  Wied.  Ann.  14,  1  (1881);  (2)  tJber  die 
Reibungskonstante  des  Quecksilberdampfes  und  deren  Abhangigkeit 
von  der  Temperatur.  Wied.  Ann.  19,  857  (1883);  15  pp. 

KLAUDY,  J.  Werth  eines  Schmiermittels.  Jahresber.  der  chem.  Tech- 
nologic. 45,  1088  (1899);  2  pp. 

KLEINHANS,  K.  The  Dependence  of  the  Plasticity  of  Rock  Salt  on  the 
Surrounding  Medium.  Physik.  Z.  15,  362  (1914);  1  p. 

KLEINT,  F.  Innere  Reibung  binarer  Mischungen  zwischen  Wasserstoff, 
Sauerstoff,  und  Stickstoff.  Verh.  D.  physik.  Gesell.  7,  146  (1905). 
Diss.  Halle  (1904). 

KLEMENCIC,  I.  .(1)  Beobachtungen  iiber  die  elastische  Nachwirkung  am 
Glase.  Carl's  Repert.  Exp.-physik.  15,  409  (1879);  18  pp.;  (2)  Beitrag 
zur  Kenntniss  der  inneren  Reibung  im  Eisen.  Carl's  Repert.  Exp.- 
physik.  15,  593  (1879);  7 pp.;  (3)  tlber  die  Dampfung  der  Schwingungen 
fester  Korper  in  Fliissigkeiten.  Wien.  Sitzungsber.  (2A),  84,  146 
(1881);  22  pp. 

KLING.  La  viscosite"  dans  ses  rapports  avec  la  constitution  chimique. 
Rev.  gen.  sci.  Paris  17,  271  (1906);  6  pp. 

VAN  KLOOSTER,  H.  S.  Normal  and  Abnormal  Cases  of  Specific  Volume  of 
Binary  Liquid  Mixtures.  J.  Am.  Chem.  Soc.  35,  145  (1913);  5  pp. 

KNIBBS,  G.  18,  19,  20,  25,  26,  34,  58,  127,  (1)  The  History,  Theory,  and 
Determination  of  the  Viscosity  of  Water  by  the  Efflux  Method.  J. 
Proc.  Roy.  Soc.  N.  S.  W.  29,  77  (1895);  70  pp.;  (2)  Note  on  Recent 
Determinations  of  the  Viscosity  of  Water  by  the  Efflux  Method.  J. 
Proc.  Roy.  Soc.  N.  S.  W.  30,  186  (1896);  8  pp. 

KNIETSCH,  R.  t)ber  die  Schwefelsaure  und  ihre  Fabrication  nach  der 
Contactverfahren,  Ber.  34,  4113  (1901);  2  pp. 


INDEX  389 

KNUDSEN,  M.  The  Law  of  Molecular  Flow  and  Viscosity  of  Gases  Moving 
through  Tubes.  Ann.  Physik.  (4)  28,  75  (1909);  55  pp.;  Polemical. 
Physic.  Rev.  31,  586  (1910;;  2  pp.  Cp.  Fisher. 

KNUDSEN,  M.  &  WEBER,  S.  Luftwiderstand  gegen  die  langsame  Bewegung 
kleinen  Kugeln.  Ann.  Phys.  36,  981  (1911);  14  pp. 

KOBLER,  B.  Untersuchungen  iiber  Viskositat  und  Oberflaschenspannung 
der  Milch.  Diss.  Zurich  (1908);  72  pp. 

KOCH,  K.  R.  Uber  die  aiissere  Reibung  tropfbarer  Fliissigkeiten.  Ann. 
Phys.  36,  613  (1911);  4  pp. 

KOCH,  S.  128,  (1)  Uber  die  Abhangigkeit  der  Reibungskonstanten  des 
Quecksilbers  von  der  Temperatur.  Wied.  Ann.  14,  1  (1881);  (2) 
Uber  die  Reibungskonstante  des  Quecksilberdampfes  und  deren 
Abhangigkeit  von  der  Temperatur.  Do.,  19,  857  (1883). 

KONIG,  W.  6,  34,  (1)  Bestimmung  einiger  Reibungscoefficienten  und 
Versuche  iiber  den  Einfluss  der  Magnetisirung  und  Elektrisirung  auf 
die  Reibung  der  Flussigkeit.  Wied.  Ann.  26,  618  (1885);  8  pp.;  (2) 
Uber  die  Bestimmung  von  Reibungscoefficienten  tropbarer  Fliissig- 
keiten mittelst  drehender  Schwingungen.  Wied.  Ann.  32,  193  (1887). 

KOLLER,  H.  Uber  den  elektrischen  Widerstand  von  Isolatoren.  Wien. 
Sitzungsber.  (2A)  98,  894  (1889;;  15  pp. 

KOHL.  Viscometer  nach  Engler  mit  konstanter  Niveau.  Z.  f.  Chem. 
App.-kunde  3,  342  (1908). 

KOHLRAUSCH,  F.  237,  (1)  Uber  die  elastische  Nachwirkung  bei  der  Torsion. 
Pogg.  Ann.  119,  337  (1863);  32  pp.;  (2)  Beitrage  zur  Kenntniss  der 
elastischen  Nachwirkung.  Pogg.  Ann.  128,  1  (1866);  20  pp.;  (3) 
Do.  Pogg.  Ann.  128,  207  (1866);  21  pp.;  (4)  Do.,  Pogg.  Ann.  128, 
339  (1866);  21  pp.;  (5)  Bemerkungen  zu  Hrn.  Neesen's  Beobachtungen 
iiber  die  elastische  Nachwirkung.  Pogg.  Ann.  165,  579  (1875);  9 
pp.;  (6)  Experimental-Untersuchungen  iiber  die  elastische  Nach- 
wirkung bei  der  Torsion,  Ausdehnung,  und  Biegung.  Pogg.  Ann. 
158,  337  (1876);  39  pp.;  (7)  Beitrag  zu  Boltzmann's  Theorie  der  elas- 
tischen Nachwirkung.  Pogg.  Ann.  160,  225  (1877);  14  pp.;  (8,  Uber 
den  Temperatureinfluss  auf  das  elektrische  Leitvermogen  von  Losungen, 
in  besondere  die  Beweglichkeit  der  einzelnen  lonen  im  Wasser.  Sig- 
zungsber.  Berlin  Akad.  1028  (1901);  8  pp.;  (9)  The  Resistance  of  the 
Ions  and  the  Mechanical  Resistance  of  the  Solvent.  Proc.  Roy.  Soc. 
London  71,  338  (1903);  13  pp.;  (10)  Weitere  Untersuchungen  iiber 
Leitvermogen.  Sitzungsber.  Berlin  Akad.  572  (1902);  9  pp. 

KOOPER,  W.  Determination  of  the  Viscosity  of  Milk  to  Detect  Added 
Water.  Milchwirtsch  43,  169,  201  (1914);  17  pp. 

KOPSCH,  W.  Die  innere  Reibung  von  Wasserstoff  und  Argon  bei  niederen 
Temperaturen.  Diss.  Halle  (1909);  44  pp. 

VON  KORANYI  and  BENCE.  Viscosity  of  Blood.  Arch.  ges.  Physiol. 
(Pfliiger's)  110. 

KORN,  A.  (1)  Viscosity  and  Elastic  Impact.  Sitz.  math.-physik.  Klasse 
Akad.  Miinchen  29,  223  (1900);  (2)  Eine  mechanische  Theorie  der 
Reibung  in  kontinuierlichen  Massensystemen.  Berlin  Dummler 
(1901);  219  pp.;  (3)  Allgemeine  Losung  des  Problems  kleiner,  station- 


390  INDEX 

arer    Bewegungen    in    reibenden    Fliissigkeiten.     Rend.    Circ.    mat. 

Palermo  26,  253  (1908);  18  pp. 
KOTLMANN.     Zeitschr.  f.  klin.  Med.  69  (1910). 
KRAUS,  C.  A.     The  Relation  between  the  Conductance  and  the  Viscosity 

of  Electrolytic  Solutions  and  its  Bearing  upon  the  Theory  of  these 

Solutions.     J.  Am.  Chem.  Soc.  36,  35  (1914);  30  pp. 
KREHL.     Pathol.  Physiologic  II  Aufl.  (1898). 

KREICHGAUER.     Quoted  by  Arrhenius  as  Wied.  Ann.  26,  161  (1885;. 
KREIDL,  A.  &  LEAK,  E.     Comparative  Studies  on  the  Viscosity  of  Milk 

by  the  Determination  of  its  Capillary  Rise.     Wien.  klin.  Wochschr. 

24,  1667  (1913);  2  pp. 
KREMANN,  R.  &  EHRLIOH,  R.     Uber  die  Fortexistenz  von  Molekiilver- 

bindungen     und     Kristallwasserhydration     im     fliissigen     Zustande. 

Monatshefte  28,  831  (1907);  62  pp. 
KROTKOV,  S.     The  Viscosity  of  the  Pleural  Exudate.     Russki.  Vratch  12, 

1399  (1913);  3  pp. 
KRUGER,  F.     Viscosity  of  Anistropic  Liquids.     Physik.  Z.  14,  651  (1913); 

4pp. 
KRUMMEL,    O.    &    RUPPIN,    E.     Innere   Reibung    des    Seewassers.     Wiss. 

Meeres,  unters,  Kiel.  Abt.  Kiel  (N.  F.)  9,  27  (1906);  9  pp. 
KRUPSAY.     Zur    Frage    der    Bildsamkeit    der    Tone.     Tonindustrie-Ztg. 

32,  1289  (1908);  1  p. 
KRUSCHE,  A.     Die  Anderung  des  Coefficienten  der  inneren  Reibung  von 

Machinenolen  mit  der  Temperatur.     Zurich.  Lohbauer  (1904);  67  pp. 
KUNDIG,   H.     tlber  die  Viskositat  des  menchlichen   Blutes  bei   Schwitz- 

proceduren.     Diss.  Jena  (1903);  25  pp. 

KTJENEN,  J.     The  Diffusion-Coefficients  of  Gases  and  the  Viscosity  Coeffi- 
cients  of   Gas  Mixtures.     Verslag.    Akad.    Wetenschappen  22,   1158 

(1913);  4  pp. 

KUENEN,  J.  P.  &  VISSER,  S.  W.     A  Viscometer  for  Volatile  Liquids.     Ver- 
slag. K.  Akad.  Wetenschappen  22  (1913);  10  pp. 
KTTLLGREN,    C.     Om   sambandet   mellan   inra   friktion   och   den   kemiska 

konstitutionen.     Oefversigt   Vet.    Acad.    Forh.    (Stockholm)    53,   647 

(1896);  4  pp. 

KUNDT.     Vorl.  Exper.  Physik.     Lect.  36,  6  pp.;  Do.  84,  7  pp. 
KUNDT,  A.  &  WARBURG,  E.     244,  (1)  tJber  Reibung  und  Warmeleitung 

verdunnter  Gase.     Pogg.  Ann.  156,  337  (1875);  29  pp.;  (2)  Do.,  Pogg. 

Ann.  166,  525  (1875);  26  pp.;  (3)  Do.,  Pogg.  Ann.  166,  17?  (1875); 

35  pp.     Cp.  Warburg. 
KUNKLER,    A.     (1)    Zur    Kenntniss  der  Mineralmachinenole.     Dingler's 

polyt.  J.  274,  323  (1889);  (2)  Do.,  280,  40;  (3)  Do.,  281,  297  (1891);  (4) 

Zeitschr.  d.  Verein  d.  Ingen.  36,  633  (1892);  5  pp.;  (5)  Do.,  74  (1893); 

Dingler's  Polyt.  J.  290,  942  (1893);  (6)  Werthbestimmung  der  Schmier- 

mittel.     Jahresber.  d.  Chem.  Technologic  46,  1088  (1895);  1  p. 
KUPPPER.     237,   238,    (1 )    Recherches   expe'rimentales  sur  1'  elasticity  des 

m6taux  faites  &  1'observatoire  physique  central  de  Russie.    (1860); 


INDEX  391 

(2)  t)ber  den  Einfluss  der  Wanne  auf  die  elastische  Kraft  der  festen 
Korper   und   insbesondere   der    Metalle.     Memoires   de   1'Acade'mie. 
St.  Petersbourg.,  Sixieme  Serie,  Sciences  Mathematiques  et  physiques 
5,  233  (1853);  &  6,  397  (1857);  97  pp. 

KURNAKOV,   N.   S.  &  EPRENOR,   N.   M.     Inner  friction  of  the  Systems: 

Chloral  Water  and  Chloral-Alcohol.     Z.  physik.  Chem.  85,  401  (1914); 

17  pp.;  Do.,  J.  Russ.  Phys.  Chem.  Soc.  45,  329  (1913);  19  pp. 
KURNAKOV,  N.  S.,  PERLMUTTER,  S.  I.  and  KANOV,  F.  P.     Viscosity  of 

Binary  Systems  Containing  Stannic  or  Antimony  Chloride.     J.  Russ. 

Phys.  Chem.  Soc.  48,  1658  (1916);  5  pp. 
KURNAKOV,  N.  &  ZHEMCHUZNUI,  S.     (1)  Die  innere  Reibung  der  binaren 

Gemische.     Characteristik  der  bestimmten  Verbindung.     Z.   physik. 

Chem.  83,  481  (1913);  26  pp.;  (2)  Flow,  Pressure  &  Hardness  of  Plastic 

Substances.     J.   Russ.   Phys.   Chem.   Soc.   46,   1004   (1913);   72  pp.; 

(3)  Inner   Friction   of   Binary   Systems.     Characteristics   of   definite 
compounds.     J.  Russ.  Chem.  Soc.  44,  1964  (1912);  27  pp.;  Z.  physik. 
Chem.  83,  481   (1913);  25  pp.;   (4)  The  Viscosity  and  Hardness  of 
Plastic  Substances.     Jahrb.  Radioakt.  Electronik  11,  1  (1914);  66  pp. 

KURZ,   A.     Die  Reibungsconstante  des  Wassers.     Ein  Vorlesungsversuch 

Exner's  Repert.  Exp.  physik.  23,  567  (1887);  4  pp. 
KURZMANN,  I.     The  New  Multi-Viscometer.     Chem.  Ztg.  37,  234  (1914); 

Kolloidchem.  Beiheft  5,  427  (1914);  Diss.  Karlsruhe  (1914). 

LABATUT.  Influence  of  Hysteresis  and  Viscosity  of  Flexure  on  the  Reading 
of  Metallic  Barometers.  Bulletin  de  la  socie'te  de  statisque,  des 
Sciences  naturelles  et  des  arts  industriels  du  department  de  1'Isere. 
28,  109  (1895). 

LABY,  T.  H.  &  CARSE,  G.  A.  On  a  Relation  Between  the  Velocity  and  the 
Volume  of  the  Ions  of  Certain  Organic  Acids  and  Bases.  Proc.  Camb. 
Phil.  Soc.  13,  287  (1906);  8  pp. 

LACHMANN,  R.  (1)  Plasticity  of  Salt  Rocks.  Centr.  Min.  46  (1912);  2  pp.; 
(2)  The  Plasticity  Question.  Do.,  745  (1912);  12pp. 

LADD,  G.  E.     Clays  of  Georgia.     Ga.  Geol.  Surv.  Bull.  6a,  29  (1898);  7  pp. 

LADENBURG,  R.  6,  33,  34,  (1)  Uber  die  innere  Reibung  zaher  Fliissigkeiten 
und  ihre  Abhangigkeit  vom  Druck.  Diss.  Miinchen  (1906);  Ann. 
Physic.  (4)  22,  287  (1907);  23  pp.;  (2)  Uber  den  Einfluss  der  Reibung 
auf  die  Schwingungen  einer  mit  Fliissigkeit  gefullten  Kugel.  Ann. 
Physik.  (4)  27,  157  (1908);  29  pp.;  (3)  Uber  den  Einfluss  von  Wanden 
auf  die  Bewegung  einer  Kugel  in  einer  reibenden  Fliissigkeit.  Ann. 
Physik.  (4)  23,  447  (1907);  11  pp. 

LAMANSKY,  S.  (1)  Untersuchungen  iiber  Schmierole.  Dingler's  Poly  tech. 
J.  248,  29  (1883);  7  pp.;  (2)  Untersuchungen  iiber  Schmierole.  Ding- 
ler's Poly  tech.  J.  256,  176  (1885);  13  pp. 

LAMB,  H.  Hydrodynamics,  A  Treatise  on  the  Mathematical  Theory  of  the 
Motion  of  Fluids.  Univ.  Press.  Cambridge  597  (1879);  223  pp. 

LAMBERT.  1,  Sur  les  fluides  considers  relativement  &  1'hydrodynamique. 
Memoires  de  1'Acad.  de  Berlin  (1784). 


392  INDEX 

LAMPE,    C.     Programm   des   stadtischen    Gymnasiums   in    Danzig  (1866). 

Schriften  der  naturforschenden  Gesell.  zu  Danzig  (N.  F.)  3,  (1872). 
LAMPEL,    A.     tlber   Drehschwingungen   einer   Kugel   mit  Luftwiderstand. 

Wien.  Ber.  (2A)  93,  291   (1886);  23  pp. 
VON  LANG,  V.     (1)  Versuche  iiber  Einstromung  von  Gasen.     Wien  Sitzungs- 

ber.  (2A)  63,  604  (1871);  15  pp.;  (2)  Zur  dynamischen  Theorie  der  Gasc. 

Wien.  Sitiungsber.     (2A)  64,  485  (1871);  4  pp.;  (3)   Do.,  Wien.  Sit- 

zungsber.  (2A)  65,  415  (1872);  4  pp.;  (4)  Experiments  on  the  Friction 

between  Water  and  Air.     Phil.  Trans.  London  166B,  589  (1876);  17  pp. 
LANSA,  L.   &   VERGANO,    R.     Iodine   Preparations   and   Viscosity   of    the 

Blood.     Pensiero    med.    No.    22-23.     Zentr.    Biochem.    Biophys.    14, 

602  (1912). 

LAQUEUR,  E.  &  SACKUR,  O.     Hofmeisters  Beitr.  3,  193  (1903). 
LASCHE,  O.     268,  Die  Reibungsverhaltnisse  in  Lagern  mit  hoher  Unfangs- 

geschwindigkeit.     Traction   &    Transmission,    Jan.,    1903);  p.   80;  Z. 

d.  Ver.  d.  Ingen.  46,  1932,  1961  (1902);  Do.     Mitt,  iiber  Forschungs- 

arbeiten.  Springer.  Heft  9  (1903);  27  pp. 
LAUENSTEIN,    C.     Untersuchungen    iiber    die   innere    Reibung   wasseriger 

Natronsalzlosungen   organischer   Sauren.     Z.    physik.    Chem.    9,   417 

(1892);  18pp. 
LAUER,  L.   &  TAMMANN,   G.     t)ber  Verschiebungselastizitat  bei  Flussig- 

keiten.     Z.  physik.  Chem.  63,  141  (1908);  10  pp.     Cp.  Tammann. 
LAWACZECK,    F.     Viscosity   and    Its    Measurement.     Z.    Ver.    deut.    Ing. 

63,  677  (1919);  6  pp. 
LEBAS,    G.     The    Theory   of    Molecular    Volumes    I.     The   Existence    of 

Additive  Relations  in  Molecular  Volumes.     Phil.  Mag.  27,  344  (1914); 

12pp. 
LE  CHATELIER,  H.  and  F.     Mechanical  Properties  of  Plastic  Substances 

(steel,  glass).     Compt.  rend.  171,  695  (1920);  4  pp. 
LECHNER,  G.     Turbulence  in  the  Flowing  of  Water  and  Mercury  through  a 

Spirally  Wound  Capillary  Tube.     Ann.  Phys.  42,  614  (1914);  28  pp. 
LEES,  C.     82,  On  the  Viscosities  of  Mixtures  of  Liquids  and  Solutions. 

Proc.  Phys.  Soc.  17,  460  (1900);  20  pp.;  Phil.  Mag.  (6)  1,  128  (1901); 

20pp. 

LEHMANN,  O.     96,  Liquid  crystals.     Physik.  Z.  7»  578  (1906);  1  p. 
LEMCKE.     Elektrische  Leitvermogen  und  innere  Reibung.     J.  Russ.  Phys. 

Chem.  Soc.  37,  1134  (1906). 
LEMSTROM,   S.     tlber  das  Verhalten  der  Fliissigkeiten  in   Capillarrohren 

unter  Einfluss  eines  elektrischen  Luftstromes.     Ann.   Physik.    (4)  6, 

729  (1901);  27  pp. 
LEPPLA,   G.     Die  Bildsamkeit   (Plastizitat)   des  Thones.     Baumaterialen- 

kunde9,  124  (1904);  2  pp. 
LEVI.     213,  Contribute  allo  studio  della  dissociazione  in  soluzioni  colloidoli. 

Gaz.  chim.  italiana  30,  64  (1900). 
LEVITES.     (1)  Sur  la  friction  inte>ieure  des  solutions  colloidales.     J.  Russ. 

Phys.  Chem.  Soc.  36,  253  (1903);  10  pp.;  Do.  36,  401  (1904);  16  pp.; 

(2)    Beitrage   zur   Kenntnis   der  Gelatinierungsvorganges.     tTber  die 

innere  Reibung  kolloiden  Losungen.     Kolloid.-Z.  2,  210  (1907);  8   pp. 


INDEX  393 

LEWY,  B.     286,  (1)  Die  Reibung  des  Blutes.     Pfliigers  Arch.  65,  447  (1897); 

26  pp.;  (2)  tlber  die  Reibung  des  Blutes  in  engen  Rohren  und  ihren 

Einfluss    auf    das    Gefalle    im    Gefasssystem.     Arch.    f.    Physiol.  147 

(1879);  3  pp. 
LEYSIEFFER,      G.     (1)     Viscosity     of     Cellulosenitrate     Solutions.     Diss. 

Berlin  (1917);  Kolloidch.  Beihefte  10,  #5  (1918);  (2)  Relation  between 

Viscosity  of  Cellulose  Nitrate  Solutions  and  the  Nitration  Process. 

Koll.  Chem.  Beihefte  10, 145  (1918);  33  pp. 
LICHTWITZ,  L.   &  RENNER,   A.     Variation  of  the  "Gold  Value"  and  the 

Viscosity  of  Colloidal  Solutions  with  the  Temperature.     Z.  Physiol. 

Chem.  92,  113  (1915);  5pp. 
LIDSTROM,  F.  M.     (1)  A  Mercurial  Viscometer.     J.  Ind.  Eng.  Chem.  36, 

270,   317  (1917);   3   pp.;    (2)    A   Modified   Mercurial   Viscometer  for 

Determining  the  Viscosity  of  Volatile  Liquids.     J.  Ind.  Eng.  Chem. 

37,  148  (1918). 

LIEBERMANN,  L.  V.     Apparatus  for  the  Determination  of  Viscosity  Particu- 
larly   that    of    Serum  and  other  Animal  Liquids.     Biochem.   Z.   33, 

218  (1911);  4  pp. 
LINDMANN,  K.     Klinische  und  experimented  Beitrage  zur  pharmakolog- 

ischen  Beeinflussung  der  Blutviscositat.     Diss.  Marburg  (1908) ;  24  pp. 
LINEBARGER,  C.     82,  90,  92,  On  the  Viscosity  of  Mixtures  of  Liquids.     Am. 

J.  Sci.  (4)  2,  331  (1896);  10pp. 
LISBONNE,  M.  &  MARGAROT,  J.     Viscosity  of  Blood.     Arch.  Malad.  coeur 

6,  279,  330  (1912;. 
LOEB,  J.     (1)  Volumetric  Analysis  of  Ion-Protein  Compounds.     Proc.  Soc. 

Exp.  Biol.   Med.  16,  39   (1918);   (2)  Amphoetric  Colloids.     IV.  The 

Influence  of  the  Valency  of  Cations  upon  the  Physical  Properties  of 

Gelatine.     J.  Gen.  Physiology  1,  483  (1919);  22pp.;  (3)  Do.     V.  The 

Influence  of  Anions  upon  the  Physical  Properties  of  Gelatine.     Do. 

1,  559  (1919).     (4)  Do.,  3,  247,  391,  547,  557,  667,  691,  827  (1920); 

142  pp.  (5)  Do.,  4,  73,  97,  187  (1921);  66  pp. 

LOWENSTEIN,  A.     Viscosity  of  the  Eye  Fluids  under  Normal  and  Patho- 
logical Conditions.     Arch.  Augenheilk.  70,  27  (1913).     Zentr.  Biochem. 

Biophys.  12,  535. 
LOWENTHAL,  J.     (1)  Die  Transpiration  der  Flussigkeiten  als   Hulfsmittel 

fiir  die  Wissenschaft  und  Technik.     Z.  anal.  Chem.  10,  298  (1871);  8 

pp.;  Do.     Z.  anal.  Chem.  11,  43  (1872);  3  pp. 
LOEWINSON  &  LESSING.     Eine  mogliche  Beziehung  zwischen  Viskositats- 

kurven  und  Molekularvolumena  bei  Silikaten.     Centralblat  Min.  289 

(1906);  2  pp. 
LOHR,    E.     Ein    einfacher   Zusammenhang   zwischen    Brechungsexponent, 

Zahigkeit,  und  Dichte  bei  Gasen.     Wien.  Ber.  (2A)  116,  1281  (1907); 

7pp. 
LOMMEL,  F.     liber  die  Viskositat  des  menschlichen  Blutes  bei  Schwitz- 

proceduren.     Deutch.  Arch.  f.  klin.  med.  80,  308  (19 — ). 
LORENTZ.     Verlagen   der  Akademie  Van   Wittenschappen  te  Amsterdam 

6,  28  (1897). 
LORENZ,  R.  &  KALMUS,  H.     Die  Bestimmung  der  inneren  Reibung  einiger 


394  INDEX 

geschmolzene  Salze.     Z.  physik.  Chem.  59,  217   (1907);    8  pp.     Cp. 

Goodwin  &  Kalmus,  and  Kalmus. 
LORIA,  G.     Viscosity  of  the  Blood  under  the  Action  of  Various  Diuretics. 

Riv.  crit.  chim.  med.  12,  #5,  6,  7;  Zentr.  Biochem.  Biophys.  13,  174. 
LOVE.     Enzyklopodie  der  Mathematischen.     Wissenshaft.  4,  II,  80;  4,  III, 

64  (1901-1908). 
Lucius,  F.     (1)  Uber  Farbstoffabsorption.     I.  Kryoskopie  und  Viskositat 

der  Milch.     II.  Diss.  Leipzig.  (1906);  54  pp. 
LUDWIG,  C.  &  STEPHAN,  J.     Uber  den  Druck,  den  das  fliessende  Wasser 

senkrecht  zu  seiner  Stromrichtung  austibt.     Wien.     Sitzungsber.  (2A) 

32,  25  (1858);  18  pp.     Cp.  Stephan. 
LUDWIK,  P.     Uber  die  Anderung  der  inneren  Reibung  der  Metalle  mit  der 

Temperatur.     Zeitschr.  physik.  Chem.  91,  232  (1916);  15  pp. 
LUDEKING,    C.     197,   213,   Leitungsfahigkeit   gelatinehaltiger   Zinkvitriol- 

losungen.     Wied.  Ann.  37,  172  (1889);  5  pp. 
LUERS,  H.  and  OSTWALD,  Wo.     Colloidal  Chemistry  of  Bread.     II.  Visco- 

metry  of  Flour.     Kolloid-Z.  25,  82,  116  (1919);  29  pp.     Cp.  Ostwald. 
LUERS,  H.  and  SCHNEIDER,  M.     289,  The  Viscosity-Concentration  Function 

of  Polydispersed  Systems.     Kolloid-Z.  27,  273  (1920);  5  pp.; 
LUNGE,   G.     (1)   Zur  Untersuchung  der  Zahfliissigkeit  von  Schmiermate- 

rialien  und  dgl.     Z.  angew.  Chem.  189  (1895);  3  pp.;  (2)  Examen  de  la 

qualite  dans  les  gommes  adragantes  au  moyen  d'un  viscosimetre  tres 

simple.     Bull.  soc.  ind.  Mulhouse  66,  64  (1896);  9  pp. 
LUNGE,  G.  &  ZILCHERT,  P.     Untersuchung  der  Zahflussigkeit  von  Gummi 

und    Traganthlosungen    mittels    des    Lungdschen    Viscosimeters.     Z. 

angew.  Chem.  437  (1895);  3  pp. 

LUSSANA,  F.     Sulla  viscosita  del  latte.     Bologna  (1905). 
LUSSANNA,  S.  &  CINELLI,  M.     L'attrito  interno  e  1'attrito  elettrolitico  nelle 

soluzioni.     Atti  d.  R.  Academia  dei  Fisiocritici  Siena  (4)  9,  49  (1897); 

17pp. 
LUST,  F.     The  Viscosity  of  the  Blood  in  Healthy  and  Sick  Infants.     Arch. 

Kinderheilk.  54,  260  (1911);  19  pp. 
LUTHE,  W.     Ballistische  Messungen  der  Magnetischen  Viskositat  an  Ringen 

aus  Elektrolyleisen,  Kobalt,  und  Nickel.     Diss.  Halle  (1912);  54  pp.; 

Ber.  physik.  Ges.  458  (1913);  28  pp. 
LYLE,  T.   &  HOSKING,   R.     The  Temperature  Variation  of  the  Specific 

Molecular   Conductivity    and    of   the    Fluidity    of    Sodium    Chloride 

Solutions.     Phil.  Mag.  (6)  3,  487  (1902);  11  pp.     Cp.  Hosking. 

MABERY,  C.  F.  &  MATTHEWS,  J.  H.     On  Viscosity  and  Lubrication.     J. 

Am.  Chem.  Soc.  30,  992  (1908);  10  pp. 

MACGILL,  A.     Proc.  &  Trans.  Roy.  Soc.  Canada  (2)  1,  III,  97  (1895). 
MACGASKEY.     The   Viscosity   of  the   Blood.     J.    Am.    Med.    Assoc.    #20 

(1908).     Cp.  Berlin  klin.  Wochenschr.  (1908). 
MACGREGOR,  J.     On  the  Relation  of  the  Physical  Properties  of  Aqueous 

Solutions  to  their  State  of  lonization.     Trans.  N.  S.  Inst.  Sci.  9,  219 

(1896-7);  27  pp.     Cp.  Barnes. 


INDEX  395 

MAC!NNES,  D.  A.  The  Ion  Mobilities,  Ion  Conductances,  and  the  Effect 
of  Viscosity  on  the  Conductances  of  Certain  Salts.  J.  Am.  Chem. 
Soc.  43,  1217  (1921);  10  pp. 

MACMICHAEL,  R.  F.     328. 

MAcNiDER,  G.  M.  (1)  A  Method  for  Determining  the  Value  of  Com- 
mercial Starches  for  Use  in  Cotton  Mills.  J.  Ind.  Eng.  Chem.  4,  417 
(1912);  12  pp.;  (2)  A  Practical  Method  for  Determining  the  Viscosity 
of  Starch  for  Mill  Purposes.  J.  Ind.  Eng.  Chem.  9,  597  (1917);  2  pp. 

MADELLA.  Sopra  alcune  determinazioni  d'attrito  interno  del  latte.  Le 
Stazioni  Sperimentali  agrarie  italiane  37,  383. 

MAGNUS,  G.  (1)  Uber  die  Bewegung  der  Fliissigkeiten.  Pogg.  Ann. 
80,  1  (1850);  36  pp.;  (2)  Hydraulische  Untersuchungen.  Pogg.  Ann. 
95,  1  (1855);  59  pp. 

MAHIN,  E.     Cp.  Jones  and  Mahin.     Diss.  Johns  Hopkins  (1908);  42  pp. 

MAHR,  H.  W.  Determination  of  the  Melting  Point  of  Greases  by  Means  of 
the  New  York  Testing  Laboratory  Viscometer.  J.  Ind.  Eng.  Chem. 
5,674  (1913);  1  p. 

MAIN,  J.  Note  on  some  Experiments  on  the  Viscosity  of  Ice.  Proc.  Roy. 
Soc.  London  42,  329  (1886);  2  pp. 

M  AIR,  J.  G.  Experiments  on  the  Discharge  of  Water  of  Different  Tempera- 
tures. Proc.  Inst.  of  Civil  Engineering  84,  II,  426  (1886);  12  pp. 

MALCOLM.     Phil.  Mag.  (6)  12,  508  (1906). 

MALLOCK,  A.  6,  (1)  Determination  of  the  Viscosity  of  Water.  Proc. 
Roy.  Soc.  London  45,  126  (1888);  7pp.;  (2)  Experiments  on  Fluid 
Viscosity.  Proc.  Roy.  Soc.  London  59,  38  (1896);  2  pp.;  Phil.  Trans. 
(A)  187,  41  (1896);  16  pp. 

MALUS,  C.  (1)  Etude  de  la  viscosite  du  soufre  aux  temperatures  supe- 
rierures  a  la  temperature  du  maximum  de  viscosite.  Compt.  rend. 
130,  1708  (1900);  3  pp.;  (2)  Recherches  sur  la  viscosite  du  soufre. 
Ann.  chim.  phys.  (7)  24,  491  (1901). 

MARCUSSON,  J.  (1)  Mitt.  a.  d.  Konigl.  Materialpriifungsamt  29,  50  (1911); 
4  pp.;  (2)  Chem.  Rev.  45  (1909).  (The  use  of  blown  oils  in  lubrication). 

MAREY.  Changements  de  direction  et  de  vitesse  d'un  courant  d'air  qui 
rencontre  des  corps  de  formes  diverses.  Compt.  rend.  132,  1291 
(1901);  5  pp. 

MARGULES,  M.  29,  Uber  die  Bestimmung  des  Reibungs-  und  Gleitungs- 
coefficienten  aus  ebenen  Bewegung  einer  Fliissigkeit.  Wien.  Sitz- 
ungsber.  (2A)  83,  588  (1881);  (2)  Wien.  Sitzungsber.  (2A)  84,  491 
(1881). 

MARIE,  C.  Surtension  et  viscosite.  Compt,  rend.  147,  1400  (1908); 
2pp. 

MARIOST.  A  Liquid  Passing  through  Another  without  Mixing.  Mem. 
soc.  sci.  phys.  nat.  Bordeau  2,  51  (1886). 

MARIOTTE.     1,  Traite  du  mouvement  des  eaux.     Paris  (1700). 

MARKOWSKI,  H.  Die  innere  Reibung  von  Sauerstoff,  Wasserstoff,  chemis- 
chem  und  atmospharischem  Stickstoff  und  ihre  Anderung  mit  der 
Temperatur.  Ann.  Physik.  (5)  14,  742  (1904);  13  pp.;  Diss.  Halle; 
41  pp.  Cp.  Bestelmeyer. 


396  INDEX 

MARKWELL,  E.     Coefficient  of  Viscosity  of  Air  by  the  Capillary  Tube 

Method.     Phys.  Rev.  8,  479  (1916);  5  pp. 
MARTENS,  A.     "Uber  die  Bestimmung  des  Fliissigkeitsgrades  von  Schmierol. 

Mitt,  der  techn.  Versuchsanst  8,  143  (1890);  8  pp. 
MARTICI,  A.     210,  Contribute  alia  conoscenza  delle  emulsioni.     Arch.  d. 

Fisiol.  4,  133  (1907). 
MARTIN,  H.  M.     Lubrication.     Proc.  Phys.  Soc.  London,  II,  32,  11  (1919); 

4pp. 
MARTINS,  F.     Die  Erschopfung  und  Ernaherung  des  Froschherzens.     Arch. 

f.  (Anat.)  u.  Phys.  543  (1882). 
MASI,  N.     Le  nuove  vedute  nelle  ricerche  teoriche  ed  esperimentali  sull' 

attrito  und  esperenze  d'attrito.     Zanichelli,  Bologna  (1897). 
MASSON.     A  Preliminary  Note  on  the  Effect  of  Viscosity  on  the  Conduc- 
tivity of  Solutions.     Austr.  Assoc.  Adv.  Sci.  3  (1901). 
MASSON,   I.   &   McCALL,  R.     Viscosity  of  Solutions  of  Nitrocellulose  in 

Mixtures  of  Acetone  and  Water.     J.  Chem.  Soc.  117,  118,  819  (1920). 
MASSOULIER,  P.     197,  (1)  Relations  entre  la  conductibilite  electrolytique 

et  le   frottement   interne   dans  les   solutions   salines.     Compt.    rend. 

130,  773  (1900);  2  pp.;  (2)  Relation  qui  existe  entre  le  resistance  e"lec- 

trique   et   la  viscosite"    des   solutions   electrolytiques.     Compt.    rend. 

143,  218  (1906);  2  pp. 
MASTROBUONO.     Viscosity  of  the  Aqueous  Humor.     Arch.   Ottalm.-Centr. 

Augenheilk.     Erganzungsheft    (1908). 
MATHIEU,  E.     14,  Sur  le  mouvement  des  liquides  dans  l«s  tubes  de  tres- 

petit  diametre.     Compt.  rend.  67,  320  (1863);  5  pp. 
MATTHEWS,  BRANDER.     7. 
MATZDORFF,  O.     A  New  Viscometer  for  the  Comparison  of  Hot,  Pasty 

Substances.     Z.  Spiritusind.  33,  420  (1910). 
MAXWELL,  J.     2,  6,  6,  128,  162,  216,  241,  243,  246,  251,  Constitution  of 

Bodies.     Encyclopedia  Brittanica.     Cp.  Theory  of  Heat;   (2)  On  the 

Dynamical  Theory  of  Gases.     Report  Brit.  Assoc.  (Pt.  2)  9  (1859);  1 

p.;  (3)  Illustrations  of  the  Dynamical  Theory  of  Gases.     Part  I.  On 

the  Motions  and  Collisions  of  Perfectly  Elastic  Spheres.     Phil.  Mag. 

(4)  19,  9  (1860);  (4)  Do.,    Part  II.  On  the  Process  of  Diffusion  of  Two 

or  More  Kinds  of  Moving  Particles  among  One  Another.     Part  III. 

On  the  Collisions  of  Perfectly  Elastic  Bodies  of  any  Form.     Phil.  Mag. 

(4)  20,  21  (1860);  (5)  On  the  Internal  Friction  of  Air  and  Other  Gases. 

Phil.  Trans.  London  156,  249  (1886);  20  pp.;  (6)  On  the  Dynamical 

Theory  of   Gases.     Phil.    Mag.    (4)   36,    129    (1868);    (7)    Do.,    Phil. 

Mag.  (4)  36,  185  (1868);  (8)  Cp.  Collected  papers. 
MAYER,  A.     (1)  Role  de  viscosite"  dans  les  ph6nomenes  osmotiques  et  dans 

les  echanges  organiques.     Compt.  rend.   Soc.   Biol.  63,   1138   (1901); 

(2)   Etudes  viscosime'triques  sur  la  coagulation  des  albuminoides  du 

plasma    sanguin    par  la  chaleur.     Compt.   rend.   Soc.    Biol.    64,   367 

(1902);  (3)  Coefficients  de  viscosite"  du  serum  et  du  plasma  sanguins 

normaux.     Compt.  rend.  Soc.  Biol.  64,  365  (1902). 
MAYER,  A.,  SCHAEFFER,  G.  &  TERROINE,  E.     Viscosity  of  soap  solutions. 

Compt.  rend.  146,  484  (1908). 


INDEX  397 

MAYESIMA,  J.     Clinical  and  Experimental  Researches  on  the  Viscosity  of 

the  Blood.     Mitt.  Grenz.  Med.  Chir.  24,  413  (1912);  25  pp. 
McBAiN,  J.  W.     Colloid  Chemistry  of  Soap.     Dept.  Sci.  Ind.  Research, 

Brit.  Assoc.  Adv.  Sci.  Third  Report  on  Colloid  Chem.  (1920);  31  pp. 
McBAiN,  CORNISH,  and  BOWDEN.     Trans.  Chem.  Soc.  101,  2042  (1912). 
McCoNNEL,  J.  239,  (1)  On  the  Plasticity  of  Glacier  and  other  Ice.     Proc. 

Roy.  Soc.  London  44,  331  (1888);  36  pp.;  (2)  On  the  Plasticity  of  an 

Ice  Crystal.     Proc.  Roy.  Soc.  London  44,  259  (1890);  1  p.;  Proc.  Roy. 

Soc.  London  49,  323  (1891);  21  pp. 
McGiLL,  A.     Viscosity  in  Liquids  and  Instruments  for  its  Measurement. 

Trans.  Roy.  Soc.  Canada  (2)  1,  97  Sect.  III.  (1895);  7  pp.;  Canadian 

Record  of  Science  6,  155  (1896). 
MclNTOSH,    D.   &  STEELE,   B.     1,  Viscosity  and  Viscosity   Temperature 

Coefficients     of    Liquids,     Hydrochloric,     Hydrobromic,     Hydriodic, 

Hydrosulphuric  Acids  and  Phosphine.     Phil.  Trans.  (A)  205,  99  (1906); 

68  pp.;  Proc.  Roy.  Soc.  London  73,  450  (1904). 
MCKEEHAN,  L.  W.     The  terminal  velocity  of  fall  of  some  spheres  in  air  at 

reduced  pressures.     Phys.  Rev.  33,  153  (1911);  16  pp. 
McM ASTER,  L.     Cp.  Jones  and  McMaster.     Diss.  Johns  Hopkins  (1906). 
MEGGITT.     A  New  Viscometer.     J.  Soc.  Chem.  Ind.  21,  106  (1902). 
MEISSNER,  W.     328,   (1)  The  Influence  of  Errors  in  the  Dimensions  of 

Engler's  Viscometer.     Chem.  Rev.  Fett-Harz-Ind.  17, 202  (1909) ;  8  pp. ; 

(2)  Chem.  Revue  iiber  die  Fett-Harz-Industrie  17,  202  (1910);  7  pp.; 

(3)  Vergleichende  Untersuchungen  iiber  den  Englischen,   Redwood' 
schen,    u.    Sayboltschen    Zahigkeitsmesser.     Chem.    Rev.    Fett-Harz- 
Ind.  19,  9  (1912);  9  pp.;  Book,  Vienna  (1912);  (4)  Comparison  of  the 
Engler,  Redwood,  and  Saybolt  Viscometers.     Chem.  Rev.  Fett-Harz- 
Ind.  19,  30,  44  (1912);  10  pp.;  Petroleum  7,  405;   (5)  Comparative 
Examination    of    Viscometers.     Chem.    Rev.    Fett-Harz-Ind.    21,    28 
(1913);  4  pp.;  (6)  Viscosity  of  Nitrocellulose.     Moniteur  Scientifique 
79  (1915). 

MELIS-SCHIRRU,  B.  Changes  in  the  Viscometric  Coefficient  of  Human 
Blood  Serum  after  Blood-Letting.  Biochem.  terap.  sper.  4,  49  (1914); 
8pp.;  Zentr.  Biochem.  Biophys.  15,  596  (1914). 

MELLOR,  J.  W.     Clay  &  Pottery  Industries.     Griffin  &  Co.,  London  (1914). 

MENNERET,  M.  Oscillatory  and  Uniform  Motion  of  Liquids  in  Cylindrical 
Tubes.  J.  physique  (5)  1,  753  (1912);  13  pp.;  Do.  1,  797  (1912); 
7pp. 

MERCANTON,  P.  L.     Simple  Lecture  Exps.     Physik.  Z.  13,  85  (1912);  1  p. 

MERCZYNG,  H.  H.  (1)  J.  Russ.  Phys.  Chem.  Soc.  21,  29  (1889);  (2)  Uber 
die  Bewegung  von  Fliissigkeiten,  Wasser  und  Petroleum  in  weiten 
Rohren.  Wied.  Ann.  39,  312  (1890);  7  pp.;  (3)  Sur  le  mouvement  des 
liquides  a  grande  vitesse  par  conduits  tres  larges.  Compt.  rend.  144, 
70  (1907);  2  pp. 

MERRILL,  G.  P.     Non-metallic  Minerals.     Wiley  &  Sons,  p.  221  (1904). 

MERRY,  E.  &  TURNER,  W.  The  Viscosities  of  Some  Binary  Liquid  Mix- 
tures containing  Formamide.  J.  Chem.  Soc.  106,  748  (1914);  1  p. 
Cp.  English. 


398  INDEX 

MERTON,  T.  R.     The  Viscosity  and  Density  of  Caesium  Nitrate  Solutions. 

J.  Chem.  Soc.  London  97,  2460  (1910);  10  pp. 
MERVEAU,  M.  J.     Recherches  sur  le  Viscosite1,  Lons  le  Saunier.     64  (1910); 

8pp. 

DE  METZ,  G.     Rigidite  des  liquides.     Compt.  rend.  136,  604  (1903);  3  pp. 
MEYER,    J.    &    MYI,IUS,    B.     Viscosity    of    Binary   Liquid    Mixtures.     Z. 

Physik.  Chem.  95,  349  (1920);  29  pp. 
MEYER,  L.     245,   (1)  tlber  Transpiration  von  Dampfen.     Part  I.  Wied. 

Ann.  7,  497  (1879);  39  pp.;  for  Part  II  Cp.  Meyer  and  Schumann; 

for  Part  III  Cp.  Steudel;  (2)  Do.     Part  IV.  Wied.  Ann.  16,  394  (1882); 

5pp. 
MEYER,  L.  &  SCHUMANN,  O.     tlber  Transpiration  von  Dampfen.     Part 

II.  Wied.  Ann.  13, 1  (1881);  19  pp. 

MEYER,  O.  E.  2,  6,  29,  50,  79,  127,  242,  243,  251,  (1)  tlber  die  Reibung 
der  Fliissigkeiten.  Crelle's  J.  rein.  Angew.  Math.  59,  229  (1861);  75 
pp.;  (2)  Do.,  Pogg.  Ann.  113,  55  (1861);  32  pp.;  (3)  Do.,  Pogg.  Ann. 
113,  193  (1861);  46  pp.;  (4)  Do.,  Pogg.  Ann.  113,  383  (1861);  42  pp.; 
(5)  tlber  die  innere  Reibung  der  Gase.  Part  I.  Pogg.  Ann.  125,  177 
(1865);  33  pp.;  (6)  Do.,  Pogg.  Ann.  125,  401  (1865);  20  pp.;  (7) 
Do.,  Pogg.  Ann.  125,  564  (1865);  36  pp.;  (8)  tlber  die  Reibung  der 
Gase.  Pogg.  Ann.  127,  253  (1866);  29  pp.;  (9)  Do.,  Pogg.  Ann. 
127,  353  (1868);  30  pp.;  (10)  tlber  die  innere  Reibung  der  Gase.  Part 

III.  Pogg.  Ann.  143,  14  (1871);  12  pp.;  (11)  Do.,  Part  IV.  Pogg.  Ann. 
148,  1   (1873);  44  pp.;  (12)  Do.,  Part  V.  Pogg.  Ann.  148,  203  (1873); 
33   pp.;    (13)    Pendelbeobachtungen.     Pogg.    Ann.    142,    481    (1871); 
43  pp.;   (14)  Theorie  der  elastische  Nachwirkung.     Pogg.  Ann.   151, 
108     (1874);     11     pp.;     (15)     Hydraulische    Untersuchungen.     Pogg. 
Ann.  Jubelb.  1   (1874);  (6)  Bemerkung  zu  der  Abhandlung  von  Dr. 
Streintz  iiber  die  Dampfung  der  Torsionsschwingungen  von  Drahten. 
Pogg.  Ann.  154,  354  (1875);  7  pp.;  (17)  Beobachtungen  von  A.  Rosen- 
cranz  iiber  den  Einfluss  der  Temperatur  auf  der  innere  Reibung  von 
Fliissigkeiten.     Wied.  Ann.  2,  387  (1877);  20  pp.;  (18)  tlber  die  elastis- 
che Wirkung.     Wied.  Ann.  4,  249  (1878);  19  pp.;  (19)  tlber  die  Bestim- 
mung  der  inneren  Reibung  nach  Coulomb's  Verfahren.     Wied.  Ann. 
32,  642  (1887);  7  pp.;  Sigzungsber.  Bayr.  Acad.  17,  343  (1887);  21 
pp.;    (20)   Ein  Verfahren  zur  Bestimmung  der  inneren  Reibung  von 
Fliissigkeiten.     Wied.  Ann.  43,  1   (1891;;  14  pp.;   (21)  De  Gasorum 
Theoria.     Diss.  Uratislaviae  (1866);  15  pp.;  (22)  tlber  die  Bestimmung 
der  Luftreibung    aus   Schwingungsbeobachtungen.     Carl's   Repert   f. 
Exper.  Physik.  18,  1  (1882);  8  pp.;  (23)  The  Kinetic  Theory  of  Gases. 
Longman  &  Co.   (1899);  466  pp.;  (24)  tlber  die  pendelnde  Bewegung 
einer  Kugel  unter  dem  Einflusse  der  inneren  Reibung  des  ungebenden 
Mediums.     J.  f.  du  reine  und  ungewandte  Math.  73,  1  (1870);  40  pp.; 
(25)   tlber  die  Bewegung  einer  Pendelkugel  in  der  Luft.     Do.,  336 
(1872);  12  pp. 

MEYER,  O.  &  ROSENCRANZ,  A.  6,  93,  127,  134.  Cp.  Meyer,  Wied.  Ann. 
2,  387  (1877). 


INDEX  399 

MEYER,  O.  &  SPRINGMUHL,  F.     tJber  die  innere  Reibung  der  Gase.     VI. 

Pogg.  Ann.  148,  526  (1873);  30  pp. 
MEYER,   P.  ••  Apparatus  for  determining   the  viscosity  of  liquids.     Ger. 

Pat.  244,098,  June  16  (1911). 
MICHAELIS,    G.     Uber  die  Theorie  der  elastischen  Nachwirkung.     Wied. 

Ann.  17,  726  (1882);  11  pp.;  The  Viscosity  of  Protein  Sols.     Biochem. 

Z.  28,  354  (1911). 

MICHAELIS,  L.  &  MOSTYNSKI,  B.     Viscosity  of  Protein  Solutions.     Bio- 
chem. Z.  25,  401  (1910);  11  pp. 
MICHELL,  A.  G.  M.     264,  268,  The  Lubrication  of  Plane  Surfaces.     Zs. 

f.  Math.  u.  Phys.  62,  123  (1905);  15  pp. 
•MiE,  G.     Remarks  upon  the  Work  of  U.  Sorkau  upon  Turbulence  Viscosity. 

Physik.  Z.  14,  93  (1913);  3  pp. 
MIFKA,  V.     The  Internal  Friction  of  Colloidal  Metal  Solutions.     Chem. 

Ztg.  35,  842  (1912). 
MILCH,  L.     The  Increase  of  Plasticity  of  Crystals  with  Rise  of  Temp. 

Neu.  Jahrb.  Min.  Geol.  Pol.  1,  60  (1909);  12  pp. 
MILLIKAN,  R.  A.     188,  242,  253,  (1)  A  New  Modification  of  the  Cloud 

Method  of  Determining  the  Elementary  Electrical  Change  and  the 

most  Probable  Value  of  that  Charge.     Phil.  Mag.  (6)  19,  215  (1910); 

20  pp.;  (2)  Most  Probable  Value  of  the  Coefficient  of  Viscosity  of  the 

Air.     Ann.  Physik.  41,  759-66  (1913). 
MILORADOV,  A.  A.  &  TOLMACHEV.     Viscosity  of  Asphalt.     J.  Russ.  Phys. 

Chem.  Soc.  Phys.  Pt.  44,  505  (1913);  8  pp. 
MINNEMANN,  J.     Note  on  Restoration  of  Plasticity  to  Pottery  Scrap  Clay. 

Trans.  Am.  Ceram.  Soc.  16,  96  (1914). 
v.    MISES,    R.     Elemente   der   Technical    Hydrodynamik.     Phys.    Z.    12, 

812  (1911). 
MOLES,  E.,  MARQUINA,  M.  &  SANTOS,  G.     Viscosidad  y  conductibilidad 

electrica   en   soluciones   concentradas   de   FeCU.     Anales   soc.    espan. 

fis.  y.  quim.  11,  Pt.  I,  192  (1913). 
MOLIN,  E.     (1)  Calculation  of  Degree  of  Viscosity  of  Mineral  Oil  Mixtures. 

Chem.  Ztg.  38,  857  (1914);  2  pp.;  (2)  Examination  of  Searle's  Method 

for  Determining  the  Viscosity  of  very  Viscous  Liquids.    Proc.  Cambridge 

Soc.  1,20,23  (1920);  12pp. 
MONROE.     105,  Cp.  Kendall  and  Monroe. 
MONSTROV,    S.     (1)    Study   of   Substances   Having  Large   Coefficients   of 

Viscosity.     VI.    Determination    of    some    Mechanical    Properties    of 

Asphalt.     J.  Russ.  Phys.  Chem.  Soc.  Phys.  Pt.  44,  492  (1913);  10  pp.; 

(2)  VII.  Supplement  to  the  Article  by  S.  I.  Monstrov.     Do.,  44,  503, 

B.  P.  Veinberg  (1913);  1  p.;  (3)  VIII.  Viscosity  of  Asphalt.     Do.,  44, 

505  (1913);  8pp. 
MONTEMARTINI,  C.     The  Relations  between  the  Water  of  Crystallization 

of  certain  Salts  and  the  Viscosity  of  their  Solutions.     Atti.  A.  Ace. 

delle  Scienze  Torino  28,  378  (1892-3);  6  pp 
MONTI,  V.     Atti.  R.  Acad.  Sci.  Torino  28,  476  (1893). 
MOORE,  B.     On  the  Viscositv  of  Certain  Salt  Solutions.     Phyic.  Rev.  3, 

321  (1896);  14  pp. 


400  INDEX 

MOORE,   H.     Valuation  of  Motor  Fuels.     Automobile  Eng.  245   (1918); 

3  pp.;  J.  Soc.  Chem.  Ind.  37,  681 A  (1918). 

MORGAN,  J.  D.     Lubricants  and  Lubrication.     Power  43,  317  (1916);  1  p. 
MORIN.     18,  Hydraulique,  45. 
MORITZ,    A.     2,    6,  Einige  Bemerkungen  iiber  Coulomb's  Verfahrendie 

Cohasion  der  Fliissigkeiten  zu  bestimmen.     Pogg.  Ann.  70,  74  (1847). 
MORRELL,    R.    S.     Varnishes,    Paints,    and    Pigments.     Dept.    Sci.    Ind. 

Research,   Brit.   Assoc.   Adv.   Sci.,   Third  Report  on   Colloid  Chem. 

(1920);  12pp. 
MORRIS-AIREY,   H.     218,  On  the  Rigidity  of  Gelatine.     Mem.   &   Proc. 

Manchester  49,  #4   (1905). 
MORTON,  W.  B.     The  Displacements  of  Particles  and  Their  Paths  in  some 

Cases  of  Two  Dimensional  Motion  of  a  Frictional  Liquid.     Proc.  Roy. 

Soc.  London  (A)  89,  106  (1913);  19  pp. 
MORUZZI,  G.     The  Effect  of  Area  on  the  Viscosity  and  Conductivity  of 

Protein  Solutions.     Biochem.  Z.  28,  97   (1911);  9  pp.;  Do.,  22,  232 

(1909). 
MOSELEY,  H.     (1)  On  the  Uniform  Flow  of  a  Liquid.     Phil.  Mag.  (4)  41, 

394  (1871);  3  pp.;  (2)  On  the  Steady  Flow  of  a  Liquid.     Phil.  Mag. 

(4)  42,  184  (1871);  14  pp.;  (3)  Do.,     Phil.  Mag.   (4)  42,  349  (1871); 

13  pp.;  (4)  Do.,     Phil.  Mag.  (4)  44,  30  (1872);  27  pp. 
MUCHIN,  G.     Fluidity  Measurements  of  Solutions.     Z.    Elektrochem.  19 

819  (1914);  2pp. 

MUELLER.     Romberg  Deutsch  Med.  Woch.  48  (1904). 
MUNZER,  E.  &  BLOCH,  F.     (1)  Die  Bestimmung  der  Viskositat  des  Blutes 

mittels  der  Apparate  von  Determann  und  Hess  nebst  Beschreibung 

eines   eigenen   Viskosimeters.     Z.    exp.    Path.    Ther.    11,   294    (1913); 

Med.   Klinik,   #9,   #10,   #11    (1909);    (2)   Experimentelle  Beitrage  zur 

Kritik  der  Viskositatsbestimmungsmethoden.     Z.  exp.  Path.  Ther.  7, 

(1909). 
MUSSEL,  A.  G.,  THOLE,  F.  B.  and  DUNSTAN,  A.  E.     The  Viscosity  of 

Compounds  Containing  Tervalent  Nitrogen.     Proc.   Chem.   Soc.   28, 

70  (1912);  J.  Chem.  Soc.  101,  1008  (1912);  8  pp.     Cp.  Dunstan. 
MUGGE,  O.     239,  Plasticity  of  Ice.     Nach.  G.  Wiss.  Gottingen.  173  (1895); 

3pp. 
MUHLENBEIN,  J.     Uber  die  innere  Reibung  von  Nichtelectrolyten.     Diss. 

Leipzig  (1901);  p.  Schettler's  Erben  (1901).     Cp.  Wagner. 
MULLER,  A.     Uber  Suspensionen  in  Medien  von  hoherer  innerer  Reibung. 

Ber.  37,  11  (1904). 
MULLER.     Studien  zur  Viscositat  des  Blutes  bei  chirurgischen  Erkrank- 

ungen.     Berlin,  klin.  Wochschr.  2276  (1909). 
MUTZEL,  K.     3,  6,  179,  Uber  innere  Reibung  von  Flussigkeiten.     Wied. 

Ann.  43,  15  (1891);  28  pp. 

NAPIERSKY.     Versuche    iiber    die    Elasticitat    der    Metalle.     Pogg.    Ann. 

Ergsbd.  3,  351  (1853). 
NATANSON,   L.    (1)    Uber   die   Gesetze   der  inneren  Reibung.     Z.  physik. 


INDEX  401 

Chem.  38,  690  (1901);  15  pp.;  Cp.  Phil.  Mag.  (G)  2,  342  (1901);  (2) 
tlber  die  temporare  Doppelbrechung  des  Lichtes  in  bewegten  reibenden 
Fliissigkeiten.  Z.  physik.  Chem.  39,  355  (1902);  9  pp.;  (3)  Uber  die 
Fortpflanzung  einer  kleinen  Bewegung  in  einer  Flussigkeit  mit  innerer 
Reibung.  Z.  physik.  Chem.  40,  581  (1902);  16  pp.;  (4)  tJber  die  Dissi- 
pationsfunction  einer  zahen  Flussigkeit.  Z.  physik.  Chem.  43,  179 
(1903);  6  pp.;  (5)  tlber  die  Deformation  einer  plastisch-viskosen 
Scheibe.  Z.  physik.  Chem.  43,  185  (1903);  18  pp.;  (6)  t)ber  einige 
von  Herrn  B.  Weinstein  zu  meiner  Theorie  der  inneren  Reibung 
gemachte  Bemerkungen.  Physik.  Z.  4,  541  (1903);  2  pp.;  Cp.  Bull. 
Int.  Acad.  Scienc.  de  Cracovie  95,  161  (1901);  Do.  19,  488,  494  (1902); 
Do.,  268,  283  (1903);  Also  Krakauer  Anz.,  95  (1901);  Do.,  488  (1902); 
Do.,  268,  283  (1903). 

NAVIER.  1,  29,  (1)  M£moire  sur  les  lois  du  mouvement  des  fluides.  Mem. 
de  1'Acad.  roy.  des  Sciences  de  Finst.  de  France  6,  389  (1823);  52  pp.; 
(2)  McSmoire  sur  l'6coulement  des  fluides  elastiques  dans  les  vases  et 
les  tuyaux  de  conduit.  M6m.  de  1'Acad.  roy.  des  Sciences  de  1'Inst. 
de  France  9,  311  (1830);  68  pp. 

NAYLOR,  R.  B.  Testing  Device  for  Determining  the  Viscosity  of  Rubber. 
U.  S.  Pat.  1,327,838,  Jan.  13  (1920). 

NEESEN,  F.  (1)  Beitrag  zur  Kenntniss  der  elastischen  Nachwirkung  bei 
Torsion.  Pogg.  Ann.  163,  498  (1874);  27  pp.;  (2)  tlber  elastische 
Nachwirkung.  Pogg.  Ann.  167,  579  (1876);  17  pp.;  (3)  Monatsber.  d. 
Kgl.  Preuss.  Acad  d.  Wissens.  (1874);  Feb. 

NENSBRUGGHE,  G.  VAN  DER.  Superficial  Viscosity  of  Films  of  Solutions  of 
Saponine.  Bull.  sci.  acad.  roy.  belg.  29,  368  (1870). 

NETTEL,  R.  Eine  neue  Viscositatsbestimmung  fur  helle  Mineralole. 
Chem.  Ztg.  29,  385  (1905);  2  pp. 

NEUFELD,  M.  W.  Influence  of  a  Magnetic  Field  on  the  Velocity  of  Flow  of 
Anistropic  Liquids  from  Capillaries.  Diss.  Danzig.  (1913);  Physik. 
Z.  14,  646  (1912);  4  pp.  Cp.  Kruger. 

NEUMANN,  F.  2,  14,  17,  (1)  Vorlesungen  iiber  die  Theorie  der  Elasticitat 
der  Festen  Korper  und  des  Lichtathers.  Leipzig.  Teubner  (1885); 
374  pp. ;  (2)  Einleitung  in  die  theoretische  Physik.  Herausgegeben  von 
C.  Pape.  Leipsig.  Teubner  (1883);  291  pp. 

NEVITT,  H.  G.  Chart  of  Viscosities  in  Different  Systems.  Chem.  Met. 
Eng.  22,  1171  (1920). 

NEWTON,  I.  1,  The  Mathematical  Principals  of  Natural  Philosophy 
(1729).  Of  the  Motion  of  Bodies.  Vol.  2.  Of  the  Motions  of  Fluids 
and  the  Resistance  Made  to  Projected  Bodies.  Section  VII. 

NEWTON,  J.  F.  and  WILLIAMS,  F.  N.  Testing  Illuminating  Oils.  Petro- 
leum Age  6,  81  (1919);  3  pp. 

NICOLARDOT,  P.  &  BAUME,  G.  A  Contribution  to  the  Study  of  the  Viscosity 
of  Lubricating  Oils.  Chimie  &  Industrie  1,  265  (1918;;  6  pp. 

NICOLARDOT,  P.  and  MASSON,  P.  J.  Dubrisay's  Method  of  Examining 
lubricating  Oils.  Ann.  fals.  II,  77  (1918);  2  pp.;  Analyst  43,  276  (1918); 

NICOLLS,   W.     Hsemodynamics.     J.  of  Physiology  20,  407  (1896). 
26 


402  INDEX 

NISHIDA,    H.     (1)    Viscosity    of    Solutions  of  Nitrocellulose  in   Alcoholic 

Solutions  of  Camphor.     Le  Caoutchouc  et  le  Gutta-Percha  121,  8103 

(1914);   (2)  Viscosity  of  Nitrocellulose  Solutions.     Kunststoffe  4,  81, 

105  (1914);  4pp. 

NISSEN.     Inaug.  Diss.  Bonn  (1880). 
NOACK,  K.     127,  (1)  Uber  den  Einfluss  der  Temperatur  und  Konzentration 

auf    die    Fluiditat   von    Flussigkeitgemischen.     Wied.    Ann.    27,    289 

(1886);  12  pp.;  (2)  tJber  die  Fluiditat  der  absoluten  und  der  verdunnter 

Essigsaure.     Wied.  Ann.  28,  666  (1886);  19  pp. 
NORDLUND,  I.     The  Validity  of  Stokes'  Law  for  the  Motion  of  Liquid  Drops 

in  other  Liquids.     Ark.  Mat.  Astron.  Fysik  9,  #13,  18  pp. 
NOYER,  G.     Viscosity  of  the  Acetates  of  Cellulose.     Caoutchouc  &  Gutta 

percha  10,  7009  (1913);  2  pp. 
NOYES,  A.  &  Al.     Conductance  and  lonization  of  Salts,  Acids,  and  Bases 

at  High  Temperatures.     Carnegie  Institution  (1908). 
NOYES,  A.  A.  &  FALK,  K.  G.     The  Properties  of  Salt  Solutions  in  Relation 

to  the  Ionic  Theory.     III.   Electrical  Conductance.     J.  Am.   Chem. 

Soc.  34,  454  (1912);  31  pp. 
NOYES,  A.  &  GOODWIN,  H.     The  Viscosity  of  Mercury  Vapor.     Physic. 

Rev.  4,  207  (1896);  10  pp.;  Z.  physik.  Chem.  21,  671  (1896);  9  pp. 
NUTTING,  P.  G.     A  New  General  Law  of  Deformation.     J.  Franklin  Inst. 

191,  679  (1921);  8  pp.;  Proc.  Am.  Soc.  Testing  Materials  (1921). 

OBERBECK,  A.  6,  Uber  die  Reibung  in  freien  Flitssighkeitsoberflachen. 
Wied.  Ann.  11,  634  (1880);  19  pp. 

v.  OBERMAYER,  A.  246,  (1)  tJber  die  Abhangigkeit  des  Reibungscoefficienten 
der  atmospharischen  Luft  von  der  Temperatur.  Wien.  Sitzungsber. 
(2A)  71,  281  (1875);  28  pp.;  Carl's  Repert.  Exp.-physik.  12,  13  (1876); 
26  pp.;  (2)  tlber  die  Abhangigkeit  des  Coefficienten  der  inneren  Reibung 
der  Gase  von  der  Temperatur.  Wien.  Sitzungsber.  (2 A)  73,  433 
(1876);  42  pp.;  Carl's  Repert.  Exp.-physik.  12,  465  (1876);  1  p.;  (3) 
Ein  Beitrag  zur  Kenntniss  der  zahfliissigen  Korper.  Wien.  Sitz- 
ungsber. (2A)  75,  665  (1977);  14  pp.;  (4)  Das  absolute  Maas  fur  die 
Zahigkeit  der  Fliissigkeiten.  Carl's  Repert.  Exp.-physik.  15,  682 
(1879);  5  pp.;  (5)  Uber  die  innere  Reibung  der  Gase.  Carl's  Repert. 
Exp.-physik.  13,  130  (1877);  29  pp.  Cp.  Wien.  Anz.  90  (1877). 

ODEN,  S.  203,  Physikalisch-chemische  Eigenschaften  der  Schwefel- 
hydrosole.  Z.  physik.  Chem.  80,  709  (1912);  38  pp. 

OEHM.  Einige  Versuche  liber  Gummilosung  als  Nahrfltissigkeit  fur  Frosch- 
herz.  Arch.  exp.  Path.  Pharm.  34,  29  (1904). 

OEHOLM,  L.  W.  189,  (1)  Free  Diffusion  of  Non-Electrolytes — the  Hydro- 
Diffusion  of  some  Organic  Substances.  Medd.  K.  Vetenskapakad. 
Nobel.  Inst.  2,  No.  23,  52  pp.;  (2)  Investigation  of  the  Diffusion  of 
some  Organic  Substances  in  Ethyl  Alcohol.  Do.  No.  24,  34  pp.;  (3) 
The  Dependence  of  the  Diffusion  on  the  Viscosity  of  the  Solvent.  Do. 
No.  26,  21  pp. 

OELSCHLAGER,  E.  The  Viscosity  of  Lubricating  Oils.  Z.  Ver.  deut.  Ing. 
62,  422  (1918);  6pp. 


INDEX  403 

OERTEL,  E.     tjbcr  die  Viscositiit  der  Milch.     Diss.  Leipzig.  (1908);  47  pp. 
OERTEL,   F.     Eine  Abanderung  der  Poiseuilleschen   Methode  zur  Unter- 

suchung  der  inneren  Reibung  in  stark  verdtinnten  wasserigen  Salz- 

losungen.     Diss.  Breslau  (1903);  48  pp. 
OHOLM,    L.    W.     (1)    Innere   Reibung   von    wasserigen   Losungen    einiger 

Nichtelektrolyten  iiber  die  Reinigung  des  hierbei  angewandte  Wassers. 

Oversight.  Finska  Vetenskaps  soc.,  Forhandlingar  47,  1  (1904;;  18  pp.; 

(2)  The  Influence  of  Non-Electrolytes  on  the  Diffusion  of  Electrolytes 
and  on  the  Electric  Conductivity,  also  a  Study  of  the  Viscosities  of  Solu- 
tions of  those  Substances.     Oversigt.     Finska  Vetenskaps.   soc.,   For- 
handlingar 65,   afd.   A.,   #5    (1913);   99   pp.     Cp.  Akad.  afh.  Helsing- 
fors  (1902). 

OFFERMANN.     Viscosity    Determinations    of    Small    Quantities    of   Oil    in 

Engler's  Viscometer.     Chem.    Rev.    Fett.-Hartz-Ind.    18,   272  (1911); 

3pp. 
ONFRAY  &  BALADOINE.     Viscosity  of  the  Blood  and  Hemorrhage  of  the 

Eye.     Soc.  ophth.  Paris,  Dec.  5  (1911);  Klin.  Monatsbl.  Augenheilk. 

13,  242. 
ONNES,  KAMERLINGH.     131,  (1)  The  Coefficients  of  Viscosity  for  Fluids  in 

Corresponding    States.     Communications    from    the    Laboratory    of 

Physics  at  the  Univ.  of  Leyden  #2;  (2)  Arch.  Ne<5rl.  30,  134  (1897);  also 

Enzykl.  d.  mathem.  Wisenschaften  V.  10,  699. 
ONNES,  K.,  DORSMAN  &  WEBER,  S.     The  Internal  friction  of  gases  at  low 

temps.     I.  Hydrogen,  II.  Helium.     Verslag.  K.  Akad.  Wetenschappen 

1375  (1913);  10  pp. 

ORR.     Proc.  Roy.  Irish  Acad.  27A,  #2  &  3  (1907). 
ORTH,  P.     Viscosity  of  Saccharin  Solutions.     Bull,  assoc.  chim.  sue.  dist. 

29,  137  (1911);  11  pp. 
ORTLOFF,    W.     Uber   die   Reibungskoefficienten    der   drei    Gasen   Aethan 

(C2H6),  Aethylen  (C2H4),  Acetylen  (C2H2).     Diss.  Jena  (1895). 
ORTON,   E.     (1)    Keram.  Rundschaw   (1901);   (2)   The  plasticity  of  clay. 

Brick  14,  216  (1901);  4  pp. 
OSEEN,  C.     (1)    Zur   Theorie  der  Bewegung  einer  reibenden  Fliissigkeit. 

Arkiv.   for  Mat.   Astron.  och  Fys.  3,  84  (1907);  (2)  Do.,  Arkiv.  Mat. 

Astron.  Fysik.  4,  1  (1908);  #9,  23  pp. 

OSMOND,  F.     Sprodigkeit  &  Plastizitat.     Paris  (1893);  8  pp. 
OST,  H.     The  Viscosity  of  Cellulose  Solutions.     Z.  angew.  Chem.  24,  1892 

(1911);  4pp.;  also  J.  Soc.  Chem.  Ind.  30,  1247  (1911). 
OSTWALD,  WM.     3,  7,  76,  76,  Physikalisch-chemische  Messungen. 
OSTWALD,  WM.  &  STEBUTT.     94,  Lehrb.  der.  allgem.  Chem.  2  auf.  22pp.; 

684  etc.  (1897). 
OSTWALD,  Wo.     206,  (1)  Zoolog.  Jahrbucher  f.  Systemat.  Geogr.  u.  Biol. 

18  (1903);  (2)  tJber  das  Zeitgesetz  des  Capillarenaufsteige  von  Flussig- 

keiten    und    die   Beziehung   derselben   zur   chemischen   Konstitution 

usw.     Z.   Chem.  Ind.   Kolloide  2,  Suppl.-heft  2,  20   (1908);   19  pp.; 

(3)  Importance   of   Viscosity   for  Study  of  Colloidal   State.     Trans. 
Faraday   Soc.    9,   34    (1913);   12pp.     Z.  f.   Chem.    u.  Ind  der  Koll. 
12,  213  (1913);  9  pp.;  (4)    Zur   Theorie    der    kritischen    Trubungen. 


404  INDEX 

Ann.  Phys.  36,  848  (1911);  7  pp.;  (5)  Pfluger's  Arch.  108,  563  (1905) ; 

(6)  Do.,  109,  277  (1905);  (7)  Do.,  Ill,  581  (1906);  (8)  Grundriss  der 

Kolloidchemie.     Theodor    Steinkopff.    Dresden.    179    (1911);  44  pp.; 

(9)  Zur  Systematik  der  Kolloide.     Kolloid.-Z.  1,  333  (1907) ;  11  pp.;  (10) 

Cp.  Liiers. 
OSTWALD,  Wo.  and  MUNDLER,  K.     The  Osmosis  and  Swelling  of  Disperse 

Systems.     Kolloid-Z.  24,  7  (1919);  20  pp. 
OSTWALD,  Wo.  u.  GENTHE,  A.     Viscosity  of  Gases  Dissolved  in  Liquids. 

Zoolog.  Jahrb.  Abt.  f.  Biol.  18, 12  (1903). 
OTT,  I.     tft>er  die  Bildung  von  Serumalbumin  in  Nagen  uber  die  Fahig- 

keit  der  Milch  das  Froschherzleistungs  fahig  zu  halten.     Arch.  (Anat.), 

Physiol.  1,  (1883). 

PACKER,  G.  Anomalia  dell'attrito  interno  dell'acqua  in  prossimita  ai 
4  gradi.  Atti.  R.  Inst.  Veneto  Scienze  68,  785  (1898);  30  pp.;  Cimento 
(4)  10,  435  (1899). 

PACKER,  G.  &  FINAZZI,  L.  35,  (1)  SulTattrito  interno  dei  liquidi  isolanti  in  un 
campo  elettrico  constante.  Atti.  R.  1st.  Veneto  di  Scienze,  II,  69, 
389  (1900);  14pp.;  (2)  Sull'attrito  interno  dell'acqua  distillata  intorno 
alia  temperatura  del  massimo  di  densita.  Atti.  R.  1st.  Veneto  di 
Scienze  58,  II,  785  (1899).  (3)  Anomalia  dell'  attrito  interno  delle 
soluzioni  acquose  in  vicinanza  alia  temperatura  del  loro  massimo  di 
densita.  Atti.  R.  1st  Veneto  di  Scienze  II,  69,  1053  (1901);  15  pp. 

PAGLIANI,  S.  (1)  Ingegnere  Civile  e  le  arti  industr.  13,  16  (1887);  (2) 
Suppl.  annuale  alia  Enciclopedia  di  Chimica  5,  (1888-9). 

PAGLIANI,  S.  &  BATELLI,  A.  Sull' Attrito  Interno  Nei  Liquidi.  Atti. 
d.  R.  Ace.  di  Torino  20,  607,  845  (1885);  28  pp. 

PAGLIANI,  S.  &  ODDONE,  E.  Sull'Attrito  Interno  Xei  Liquidi.  Atti. 
d.  R.  Ace.  di  Torino  22,  314  (1887);  9  pp. 

PAINLEVE,  P.     Lecons  sur  le  frottement.     Ill  pp. 

PARISH,  W.  High  Viscosity  Oklahoma  Oil  Versus  Low  Viscosity  of  Pennsyl- 
vania Oil.  Nat.  Petroleum  News  6,  #9,  36  (1913);  7  pp. 

PARNELL,  J.  and  HIGGINS.  National  Physical  Lab.  Collected  Researches 
13  (1916). 

PARTURIER,  M.  and  DONS-KAUFMANN,  M.  (1)  Influence  of  Digitalis  on 
the  Viscosity  of  the  Blood  in  Cardiac  Asystole.  Compt.  rend.  soc. 
biol.  80,  407  (1917);  4  pp.;  (2)  Influence  of  Potassium  Iodide  on  the 
Viscosity  of  the  Blood.  Compt.  rend.  soc.  biol.  80,  456  (1917);  3  pp. 

PATTERSON,  W.  A  Constant  Pressure  Viscometer.  Proc.  Chem.  Soc. 
29,  172  (1913). 

PAULI,  W.  188,  213,  (1;  Viskositat  und  Elektrochemie  der  Eiweisslosungen. 
Z.  Chem.  Ind.  Kolloid  12,  222  (1913);  8  pp.;  Cp.  Trans.  Faraday  Soc. 
9,54  (1913);  12pp. 

PAULI,  W.  &  WAGNER,  R.  The  Viscosity  of  Protein  Sols.  Biochem.  Z. 
27,  296  (1910);  8pp. 

PAUSCHMANN.     Diss.  Erlangen  (1910). 

PEDDIE,  W.  On  the  Toreional  Oscillations  of  Wires.  Phil.  Mag.  (5) 
38,36  (1894);  19pp. 


INDEX  405 

PEDERSON,  F.     The  Influence  of  Molecular  Structure  upon  the  Internal 

Friction  of  Certain  Isomeric  Ether  Gases.     Phys.  Rev.  25,  225  (1907). 
PELLET,  M.     Relation  between  the  Fluidity   (Barbey)  and  the  Viscosity 

(Engler)  of  Lubricating  Oils.     Bull.  Assoc.  Chem.  Sucr.  dist.  29,  622 

(1913;;  2pp. 
PERRIN.     190. 
PERROTT,    G.    ST.    J.    and   THIESSEN,    R.     218,   282,    Carbon    Black— Its 

Properties  and  Uses.     J.  Ind.  Eng.  Chem.  12,  324  (1920);  8  pp. 
PERRY,  J.,  GRAHAM,  J.  &  HEATH,  C.     6,  131,  Liquid  Friction.     Phil.  Mag. 

(5)  36,  441  (1893;;  18  pp. 

PETROFF,  N.  263,  268,  (1)  Neue  Theorie  der  Reibung.  J.  des  Ingenieurs 
Nos.  1,  2,  3  (1883);  Imp.  Russ.  Acad.  of  Sciences,  St.  Petersburg; 
(2)  Frottement  dans  les  machines.  Memoires  de  F  Academic  de  St. 
Petersburg  (8)  10,  No.  4  (1890);  (3)  Experimentale  Untersuchungen 
iiber  die  Reibung  der  Fliissigkeiten.  Petersburg  (1886);  (4)  Neue 
Theorie  der  Reibung.  Trans,  by  Wurzel.  Voss.  Leipzig  (1887); 
pp. ;  (5)  Uber  ein  physikalisches  Verfahren  zur  Bestimmung  der  Eigen- 
schaften  eines  Schmiermittels.  Baumaterialienkunde,  269  (1899); 

(6)  Prosede   de   determination   des   qualite's   d'un   liquide   lubrifiant. 
Congres    international    des  methodes  d'essai  des  matcriaux  de  con- 
struction 2,  1  (1901);  6  pp.;  Paris.  Vve.  Ch.  Dunod.;  (7)  Reibung  der 
Fliissigkeiten  und   Machinen    (russ).     Ber.   d.    St.    Peterburger  tech- 
nolog.  Instituts,  1885-6;  J.  f.  Ingenieur  (russ.),  (1883).     Nos.  1,  2,  3,  4; 
J.  d.  russ.  Phys.-chem.  Ges  16,  14  (1884);  Bull.  d.  kaiserl.  Akad.  d. 
Wissen.  zu.  St.  Petersburg  5,  365  (1896). 

PFAFF,    F.     239,   Versuche   uber   die   Plasticitat   des   Eises.     Pogg.    Ann. 

155,  169  (1875);  6  pp.;  Sitzungsber.  der  phys.  med.  Soc.  zu  Erlangen, 

72  (1875). 

PHILIP,  A.     Reducing  the  Viscosity  of  Petroleum  Oils.     U.  S.  Pat.  1,286,091. 
PHILLIPS,  P.     143,  146,  245,  Viscosity  of  Carbon  Dioxide.     Proc.  Roy.  Soc. 

London  87,  48  (1912);  13  pp. 
PICCIATI,   G.     Sul  moto  di  un  cilindro  indefinite  in  un  liquido  viscoso. 

Line.  Rend.  (5)  16,  (2)  174  (1907);  10  pp. 
PICCININI,  G.  M.     Viscometric  and  Cryoscopic  Variations  in  the  Blood  after 

Administration    of    Antipyrine,    Phenacetin    and    Antifebrin.     Arch. 

farm.  sper.  12,  193  (1911);  17  pp. 
PICK,  H.     Viscosity  of  Fluid  Crystallin  Mixtures  of  p-Azoxyanisole  and 

p-Azoxyphenetole.     Z.  physik.  Chem.  77,  577  (1911);  10  pp. 
PICKERING,  S.  U.     Uber  Emulsionen.     Koll.  Zeitschr.  7,  11  (1910). 
PIERCE,  C.     On  the  Influence  of  the  Internal  Friction  upon  the  Correction 

of  the  Length  of  the  Seconds  Pendulum  for  the  Flexibility  of  the 

Support.     Proc.  Am.  Acad.  13,  396  (1878);  6  pp. 
PIEST,  C.     (1)  The  Viscosity  of  Nitrocellulose  Solutions.     Z.  Ges.  Schiess- 

Sprengstoffw.  6,  409  (1911);  5pp.;Z.  angen.  Chem.  24, 968  (1911);  4  pp. 
PISARSHEWSKI,  L.  and  KARP,  E.     The  Relation  between  Diffusion  Constant, 

the  Viscosity  and  the  Electrical  Conductivity.     Z.  physik.  Chem.  63, 

257  (1908);  12  pp.;  Cp.  J.  Russ.  Phys.  Chem.  Soc.  Phys.  Pt.  40,  599. 


406  INDEX 

PISARSHEWSKI,  L.  &  L.EMPKE.  196,  Electroconductibilite"  et  frottement 
inte"rieur.  J.  Russ.  Phys.  Chem.  Soc.  37,  492  (1905);  10  pp. 

PISATI,  G.  6,  237,  (1)  Su  la  dilatazione,  la  capillarita  e  la  viscosita  del 
solfo  fuso.  Atti.  R.  Ace.  Lined  74,  150  (1877);  5  pp.;  (2)  Sulla  elas- 
ticita  del  metalli  a  diverse  temperature.  Gazz.  chim.  ital.  6  (1876); 
Cp.  Do.  7  (1877);  (3)  Beitrage  zur  Kenntniss  der  elastischen  Nach- 
wirkung.  Wien.  Sitzungsber.  (2A)  80,  427  (1879);  12  pp. 

PIVNIKIEVICZ,  H.  A  Simple  Apparatus  for  the  Absolute  Determination  of 
the  Viscosity  Coefficient  and  the  Demonstration  of  Maxwell's  Law. 
Physik.  Z.  14,  305  (1913);  3  pp. 

PLATEAU,  J.  Statique  experimental  et  theorique  des  liquides  soumis  aux 
seules  forces  moleculaires.  Paris,  Ganthier-Villars  (1873). 

PLEISSNER,  M.  Untersuchungen  tiber  die  relative  innere  Reibung  von 
Speisefetten  und  fetten  Olen.  Arch.  Pharm.  242,  24  (1904);  7  pp. 

PLESSI,  A.  and  VANDINE,  D.  Viscometry  of  the  Blood  in  Various  Diseases. 
Riv.  crit.  clin.  med.  12,  609  (1911);  5  pp. 

POCHETTINO,  A.  235,  Su  le  proprieta  dei  corpi  plastici.  (Viscosity  of 
Pitch).  Nuovo  cimento  8,  77  (1914);  31  pp. 

POINCARE,  L.  Recherches  sur  les  electrolytes  fondus.  Ann.  chim.  phys. 
(6)  21,  289  (315)  (1890);  2  pp. 

POISEUILLE.  1,  6,  8  et  seq.  17/66,  62,  68,  70,  72,  127,  134,  138,  178,  179, 
284,  286,  (1)  Recherches  experimentales  sur  le  mouvement  des  liquides 
dans  les  tubes  de  tres-petits  diametres.  Compt.  rend.  15,  1167  (1842); 
20  pp.;  Cp.  Ann.  chim.  phys.  (3)  7,  50  (1843);  24  pp.;  Pogg.  Ann. 
58,  424  (1843);  24  pp.;  M6m.  pro's,  par  divers  savants  a  1'acad.  Roy. 
des.  Scienc.  de  1'inst.  de  France  9,  433  (1846);  111  pp.;  Ann.  64,  129 
(1848);  (2)  Recherches  sur  les  causes  du  mouvement  du  sang  dans  les 
vaisseaux  capillaires.  Ann.  chim.  u.  Pharm.  64,  129;  (3)  Recherches 
experimentales  sur  le  mouvement  des  liquides  de  nature  diffdrente  dans 
les  tubes  de  tres-petits  diametres.  Ann.  chim.  phys.  (3)  21,  76  (1847); 
34  pp. 

POISSON.  1,  Calcul  des  pressions  dans  les  fluides  en  mouvement.  Equa- 
tions differentielles  de  ce  mouvement.  J.  de  l'e"cole  polytechn.  13, 
139  (1831);  36  pp. 

POLLOCK,  J.  A.  Relation  between  the  Thermal  Conductivity  and  the 
Viscosity  of  Gases.  J.  Proc.  N.  S.  Wales  63,  116  (1919);  3  pp. 

PONTIO,  M.  The  Viscosity  of  Rubber  Solutions.  Caoutchouc  et  Gutta- 
percha  8,  5108  (1911);  J.  Soc.  Chem.  Ind.  30,  699  (19—). 

POOLE,  C.  P.  Cylinder  Oil  Viscosity.  (Use  of  Aluminium  Oleate).  Power 
43,  885  (1916). 

PORST,  C.  E.  G.  and  MOSKOWITZ,  M.  Comparison  of  the  Various  Corn 
Products  Starches  as  Shown  by  the  Bingham-Green  Plastometer. 
J.  Ind.  Eng.  Chem.  14,  49  (1922);  4  pp. 

PORTER,  A.  W.  (1)  Notes  on  the  Viscosity  of  Liquids.  Phil.  Mag.  23, 
458  (1912);  4  pp.;  (2)  Vapor  Pressure.  Phil.  Mag.  23,  4912) ;8  (15 
5pp. 

POST.     Chem.-tech.  Analyse  1,  Heft  2,  318. 


INDEX  407 

POUND.     Physical  Properties  of  Mixtures  of  Ether  and  Sulphuric  Acid. 

J.  Chem.  Soc.  99,  708  (1911);  15  pp. 
POWELL,  C.     (1)  The  Viscosity  of  Sugar  Solutions.     J.  Chem.  Soc.  106, 

1  (1914);  23  pp.;  (2)  Determination  of  the  Viscosity  of  Refined  Syrups 

and  Molasses.     J.  Soc.  Chem.  Ind.  33,  238  (1914);  2  pp. 
POYNTING,   J.     Change  of   State:   Solid-Liquid.     Phil.    Mag.    (5)    12,   32 

(1887);  17pp. 
PRANDTL,  L.     (1)  Handworterbuch  der  Naturwissenschaften.     Jena  (1913) 

vide  Fliissigkeitsbewegung;    (2)  Neue  Untersuchungen  iiber  die  stro- 

mende  Bewegung  der  Case  und  Dampfe.     Physik.  Z.  8,  23  (1907);  9  pp. 
PREDENNING.     Viscositat   und   magnetische    Doppelbrechung  des  colloid- 

alen     Eisenoxyhydrates.     Diss.     Heidelberg     (1904);    40    pp.     Rep. 

British  Assoc.   Cambridge  476   (1904);   1  p.;  Proc.  Roy.  Soc.  (B)  78, 

328  (1906);  30  pp. 
PRESTON,  A.  C.     (1)  Flow  of  Oil  in  Pipes.     Chem.  Met.  Eng.  23,  607  (1920) ; 

6  pp.;  (2)  Do.,  do.  23,  685  (1920);  4  pp. 
PRIBRAM.     Uber    die    Beziehungen    zwischen    inneren    Reibung    und    der 

chemischen  Zussammansetzung  fliissiger  Substanzen.     Lehrbuch  der 

Chem.  Graham  Otto,  Vol.  1,  Pt.  Ill,  Chap.  3. 
PRIBRAM,  R.  &  HANDL,  A.     2,  6,  62,  63,  75,  106,  (1)  tlber  die  spezifische 

Zahigkeit  der  Fliissigkeiten  und  ihre  Beziehung  zur  chemischen  Kon- 

stitution.     Part  I.  Wien  Sitzungsber.    (2A)  78,   113   (1878);  52  pp.; 

Carl's  Repert.   Exp.-Physik.   16,  465  (1879);  (2)  Do.,  Part  II.  Wien. 

Sitzungsber.    (2A)    80,    17   (1879);  41  pp.;   (3)   Do.,  Part  III.  Wien. 

Sitzungsber.    (2A)  84,  717  (1881);  73  pp.;  (4)  Do.,  Z.  physik.  Chem. 

9,  529  (1892);  11  pp. 
PRICHARD,  H.  S.     The  Effects  of  Straining  Structural  Steel  and  Wrought 

Iron.     Proc.  Am.  Soc.  Civ.  Eng.  42,  69  (1916);  45  pp.     Discussion  in 

March  and  April  Nos. 
PRINSEN.     Die  Viscositat  der  Rohrzuckersirupen  (der  Einfluss  von  gelosten 

Salzen  und  Andern.     Holland.  Arch.  Java  Suiker  11,  3  (1903);  18  pp. 
PRONY.     1,   Recherches  physico-mathe'matiques  sur  la   the"orie   des  eaux 

courantes.     Paris  (1804). 
PROUDMAN,  J.     Notes  on  the  Motion  of  Viscous  Liquids  in  Channels.     Phil. 

Mag.  (6)  28,30  (1914);  7pp. 
PUCCIANTI,  L.     (1)  Misure  di  viscosita  sopra  i  cristalli  fluidi  delLehmann. 

Atti  accad.  Lincei  (5)  16, 1,  754  (1907);  3  pp. 
PULLEN  &  FINLEY.     Mech.  Engineering  24,  493.     Inst.  Mech.  Engineerg. 

Gt.  Brit.  2,  43  (1909). 
PULTJJ,  J.     79,  246,  261,  252,   (1)  Uber  die  Reibungsconstante  der  Luft 

als    Function    der   Temperature.     Wien.    Sitzungsber.    (2A)    69,    287 

(1874);  35  pp.;   (2)  Do.     Wiener  Sitzungsber.   (2A)  70,  243   (1875;; 

25  pp.;    (3)  tlber  die  Abhangigkeit  der  Reibung  der  Gase  von  der 

Temperatur.     Wien.   Sitzungsber.    (2A)   73,  589   (1876);  40  pp.;  Cp. 

Carl's   Repert.   Exp.-physik.   13,  293    (1877);   15  pp.;    (4)   Uber  die 

Reibung  der  Dampfe.     Wien.  Sitzungsber  (2A)  78,  279  (1878);  33  pp.; 

Cp.  Carl's  Repert.  Exp.-physik.  15,  427  (1879);  31  pp.;  (5)  Uber  die 


408  INDEX 

innere  Reibung  in  einem  Gemische  von  Kohlensaure  und  Wasserstoff. 

Wien.  Sitzungsber.  (2A)  79,  97  (1879);  17  pp.;  (6)  Do.,  Part  II.,  Wien 

Sitzungsber.  (2A)  79,  745  (1879);  12  pp. 
PtiRDY,  R.  C.     Discussion  of  paper  by  Ashley  on  Colloid  Matter  of  Clay  and 

its  Measurement.     Trans.  Am.  Ser.  Soc.  11,  555  (1909);  40  pp.;   111. 

Geol.  Surv.  9  (1908). 
PYHALA,  C.     The  Viscosity  of  Technical  Naphthenic  Acids.     Petroleum 

9,  1373  (1914);  1  p.;  Do.,  217  (1911). 
PYTNKHOV,  S.     J.  Russ.  Phys.  Chem.  Soc.  41,  665  (1909);  2  pp. 

QUARTAROLI.  Ricerche  sperimentali  sull'attrito  interne  delle  soluzioni 
colloidali  vicino  al  punto  di  massima  densita.  Forli  (Medri  e  C.) 
(1901);  14pp. 

QUINCKE,  C.  34,  Die  Klebrigkeit  isolirender  Fliissigkeiten  im  constanten 
elektrischen  Felde.  Wied.  Ann.  62,  1  (1897).  Cp.  Schaufelberger. 

RABER,  S.  Die  Konstante  der  inneren  Reibung  des  Ricinusols,  und  das 
Gesetz  ihrer  Abhangigkeit  von  der  Temperatur.  Cp.  Kahlbaum  & 
Raber.  Diss.  Halle  (1904);  107  pp. 

RAGOSINE.  t)ber  das  Viscosimeter  Engler-Ragosine.  Chem.  Ztg.  26, 
628  (1901). 

RAKKUK,  F.  237,  Die  elastische  Nachwirkung  bei  Silber,  Glas,  Kupfer, 
Gold,  Platin,  und  Zink,  inbesondere  deren  Abhangigkeit  von  der 
Temperatur.  Wied.  Ann.  35,  476  (1888);  20  pp. 

RAKUSIN.  Untersuchung  der  Erdole  und  seiner  Produkte.  Braunschweig 
(1906);  271  pp. 

RAMSAY  and  SHIELDS.     121,  123. 

RANKEN,  C.  &  TAYLOR,  W.  179, 186,  Viscosity  of  Solutions.  Trans.  Roy. 
Soc.  Edinburgh  45,  397  (1906);  9  pp.  Cp.  Taylor  &  Ranken. 

RANKINE,  A.  O.  7,  250,  251,  253,  (1)  Method  of  Determining  the  Viscosity 
of  Gases,  Especially  Those  Available  only  in  Small  Quantities.  Proc. 
Roy.  Soc.  London  A  83,  265  (1910);  11  pp.;  (2)  Viscosity  of  the  Gases  of 
the  Argon  Group.  Physik.  Z.  11,  497  (1910);  5  pp.;  (3)  The  Variation 
with  Temperature  of  the  Viscosity  of  the  Gases  of  the  Argon  group. 
Proc.  Roy.  Soc.  London  (A)  84,  181  (1910);  12  p.;  Physik.  Z.  11,  745 
(1910);  8  pp.;  (4)  Relation  between  Viscosity  and  Atomic  Weight 
for  the  Inert  Gases  with  its  Application  to  the  Case  of  Radium  Eman- 
ation. Phil.  Mag.  (6)  2,  145  (1911);  9  pp.;  (5)  Viscosities  of  Gaseous 
Chlorine  and  Bromine.  Proc.  Roy.  Soc.  London  (A)  86,  162  (1912); 
6  pp. ;  (6)  Method  of  Measuring  the  Viscosity  of  the  Vapors  of  Volatile 
Liquids  with  the  Application  to  Bromine.  Proc.  Roy.  Soc.  London  (A) 
88,  575  (1913);  13  pp.;  (7)  Phil.  Mag.  40,  516  (1920);  (8)  On  the  prox- 
imity of  atoms  in  gaseous  molecules.  Proc.  Roy.  Soc.  A  98,  360  (1921); 
10  pp. 

RANKINE,  W.  J.  M.  (1)  Thermodynamic  Acceleration  and  Retardation 
of  Streams.  Phil.  Mag.  40,  288  (1870);  (2)  Mathematical  Theory  of 
Combined  Streams.  Proc.  Roy.  Soc.  London  19,  90  (1871). 


INDEX  409 

RAPP,  I.  M.  242,  The  Flow  of  Air  through  Capillary  Tubes.  Phys.  Rev. 
2,363  (1914);  19pp. 

RAPPENECKER,  K.  Viscosity  Coefficient  of  Vapors  and  their  Variation 
with  the  Temperature.  Diss.  Freiburg  i  B.  (1909);  Z.  physik.  Chem. 
72,695  (1910);  27pp. 

RASSOW,  B.  &  DORHLE,  O.     Kolloid.-Z.  12,  Feb.  (1913). 

RAYLEIGH,  LORD.  264,  (1)  On  the  Motion  of  Solid  Bodies  through  a  Vis- 
cous Liquid.  Phil.  Mag.  (6)  21,  697  (1911);  15  pp.;  (2)  Notes  on  the 
Theory  of  Lubrication.  Phil.  Mag.  (6)  35,  1  (1918);  12  pp.;  (3)  On  the 
Lubricating  and  other  Properties  of  Thin  Oily  Films.  Phil.  Mag. 
(6)  36,  157  (1918);  6  pp.;  (4)  On  Flow  of  Viscous  Liquids,  especially 
in  Two  Dimensions.  Sci.  papers  IV  p.  78;  (5)  Viscosity  of  Argon  and 
Helium.  Sci.  papers,  IV.,  p.  222;  (6)  Viscosity  of  Hydrogen  as  Affected 
by  Moisture.  Sci.  papers,  III,  p.  375;  Proc.  Roy.  Soc.  62,  112  (1897); 
5  pp.;  (7)  On  the  Viscosity  of  Argon  as  affected  by  the  Temperature. 
Sci.  papers,  IV.,  452;  Proc.  Roy.  Soc.  66,  68  (1900);  7  pp.;  (8)  On  the 
Viscosity  of  Gases  as  affected  by  Temperature.  Do.,  p.  484;  Proc. 
Roy.  Soc.  67,  137  (1900);  3  pp.;  (9)  Foam.  Discussion  of  Superficial 
Viscosity.  Do.,  p.  351;  (10)  On  the  Superficial  Viscosity  of  Water. 
Do.,  p.  363;  (11)  Friction  and  Heat  Conduction.  Theory  of  Sound  2, 
312  Chap.  19,  and  Appendix  A;  (12)  Further  Remarks  on  the  Stability 
of  Viscous  Fluid  Motion.  Phil.  Mag.  (6;  28,  609  (1914);  11  pp.; 
(13)  On  the  Stability  of  the  Flow  of  Fluids.  Phil.  Mag.  (5)  34,  59 
(1892);  (14)  Scientific  Papers.  3,  351,  363;  4,  78,  222,  336,  375,  452 
and  481. 

REDWOOD,  B.  324,  (1)  On  Viscometry.  J.  Soc.  Chem.  Ind.  6,  121  (1886); 
11  pp.;  (2)  Action  of  Oils  on  Metals.  J.  Soc.  Chem.  Ind.  362  (1886); 
2  pp.;  (3)  Petroleum.  London,  Griffin  &  Co.  2  (1896);  873  pp.;  (4) 
Petroleum  and  Its  Products  (1906). 

REGECZY-NAGY,  E.  Stromung  von  Fliissigkeiten  in  Capillarrohren. 
Math.  Naturw.  Ber.  aus  Ungarn.  1,  232  (1882);  1  p. 

REIGER,  R.  58,  239,  (1)  Innere  Reibung  plastischer  und  fester  Korper. 
Physik.  Z.  2,  213  (1901).  Diss.  Erlangen  (1901);  5  pp.;  (2)  tlber  die 
Gultigkeit  des  Poiseuilleschen  Gesetz  bei  zahflussigen  und  festen 
Korpern.  Ann.  Physik.  (4)  19,  985  (1906);  22  pp.;  (3)  Uber  die  station- 
are  Stromung  einer  Substanz  mit  innerer  Reibung  und  den  Einfluss  der 
Elastizitat  der  Wand.  Erlanger  Ber.  38, 203  (1906) ;  15  pp.  Cp.  Glaser; 
(4)  Propagation  of  Shearing  Deformations  in  Liquids.  Ann.  Physik. 
(4)  31,  51  (1909);  41  pp.;  Erlanger  Ber.  40,  160  (1908);  15  pp. 

REINGANUM,  M.  (1)  tlber  die  Theorie  der  Zustandsgleichung  und  der 
inneren  Reibung  der  Case.  Physik.  Z.  2,  241  (1901);  5  pp.;  (2)  Varia- 
tion with  Temperature  of  the  Viscosity  of  Gases  of  the  Argon  Group. 
Physik.  Z.  12,  779  (1911). 

RELLSTAB,  L.  2,  6,  106,  Uber  die  Transpiration  homologer  Fliissigkeiten. 
Inaug.  Diss.  Bonn  (1868). 

REPIN,  C.  Experiences  du  lavage  mecanique  du  sang.  Compt.  rend.  141, 
221  (1905;. 


410  INDEX 

REYHER,  R.  3,  Uber  die  innere  Reibung  von  Losungen.  Z.  physik.  Chem. 
2,  744  (1888);  14  pp. 

REYNOLDS,  F.  (1)  The  Viscosity  Coefficient  of  Air  and  an  Inquiry  into  the 
Effect  of  Roentgen  Rays  thereon.  Physic.  Rev.  18,  419  (1904);  22  pp.; 
(2)  Do.,  Physic.  Rev.'l9,  37  (1904);  10  pp. 

REYNOLDS,  O.  18,  21,  29,  et  seq.  130,  264,  et  seq.  (1)  An  Investigation  of  the 
Circumstances  which  determine  whether  the  Motion  of  Water  shall  be 
Direct  or  Sinuous,  and  of  the  Law  of  Resistance  in  Parallel  Channels. 
Phil.  Trans.  London  174,  935  (1883);  48  pp.;  Cp.  Roy.  Inst.  of  Gt. 
Brit.  9,  44  (1884);  (2)  On  the  Theory  of  Lubrication  and  its  Application 
to  Mr.  B.  Tower's  Experiments,  including  an  Experimental  Deter- 
mination of  the  Viscosity  of  Olive  Oil.  Phil.  Trans.  London  177  A,  157 
(1886);  78  pp.;  (3)  On  the  Dynamical  Theory  of  Incompressible  Viscous 
Fluids  and  the  Determination  of  the  Criterion.  Phil.  Trans.  London 
186  A,  123  (1895);  42  pp.;  Cp.  Brit.  Assoc.  Rep.  Martical  (1884);  Phil. 
Trans.  160  (1886);  83  pp.;  Collected  Papers  2,  228  (1886). 

RIBAUCOUR,  A.  Hydrodynamic  Phenomena  in  Mixed  Turbid  and  Clean 
Water.  Compt.  rend.  252  (1885). 

RICHARDS,  T.  W.  and  PALITZSCH,  S.  Compressibility  of  Aqueous  Solutions, 
especially  of  Urethan,  and  the  Polymerization  of  Water.  J.  Am. 
Chem.  Soc.  41,  59  (1919);  10  pp. 

RICHARDSON,  CLIFFORD.  204,  205,  (1)  The  Modern  Asphalt  Pavement; 
(2)  The  Colloidal  State  of  Matter  in  Its  Relation  to  the  Asphalt 
Industry.  Dept.  Sci.  Ind.  Research.  Brit.  Assoc.  Adv.  Sci.,  Third 
Report  on  Colloid  Chem.  (1920);  5  pp. 

RIDDELL,  M.  The  Fluidity  of  Molten  Cast  Iron.  Foundry  46,  408  (1918); 
3  pp.;  Foundry  Trade  J.  20,  364  (1918);  3  pp.;  J.  Am.  Soc.  Mech.  Eng. 
40,  860  (1918);  1  p. 

RIEKE,  R.     Plasticity  of  Clay.     Ceramique  16,  87  (191-). 

RIGG,  G.  and  CARPENTER,  J.  L.  Stormer  Viscometer  and  the  Value  of 
Viscosity  Determinations  by  its  Use.  J.  Ind.  Eng.  Chem.  4,  901 
(1913);  2  pp. 

RINGER,  W.  E.  Protein  Acid  Combinations  and  Viscosity.  Van  Bem- 
melen-Festschrift,  243  (1911);  18  pp. 

RIVETT,  A.  C.  D.  &  SIDGWICK,  N.  V.  The  Rate  of  Hydration  of  Acetic 
Anhydride.  J.  Chem.  Soc.  97,  732  (1910);  9  pp. 

ROBERTS,  H.  T.  Method  of  Investigating  the  Transpiration  of  Gases 
through  Tubes.  Phil.  Mag.  23,  250  (1912);  5  pp. 

ROBERTS-AUSTEN,  W.  C.  Properties  Common  to  Fluid  and  Solid  Metals. 
Proc.  Royal  Institution  11,  395  (1887). 

ROBERTSON,  T.  B.  Notiz  iiber  einige  Faktoren,  welche  die  Bestandteile 
von  Ol-Wasser  Emulsionen  betimmen.  Kolloid-Z.  7,  7  (1910);  3  pp. 

RONTGEN,  W.  138,  (1)  Uber  einen  Vorlesungsapparat  zur  Demonstration 
des  Poiseuille'schen  Gesetz.  Wied.  Ann.  20,  268  (1883);  4  pp.;  (2) 
Uber  den  Einfluss  des  Druckes  auf  die  Viscositat  der  Fliissigkeiten, 
speciell  des  Wassers.  Wied.  Ann.  22,  510  (1884);  9  pp.;  (3)  Kurze 
Mitteilung  von  Versuchen  liber  den  Einfluss  des  Druckes  auf  einige 


INDEX  411 

physikalische  Erscheinungen.  Ann.  Phys.  Chem.  (3)  45,  98  (1892); 
10  pp.  (Viscosity  and  conductivity.) 

ROGERS,  A.  &  SABIN,  A.  H.  Consistency  of  Paints  by  the  Stormer  Visco- 
meter.  J.  Ind.  Eng.  Chem.  3,  737  (1911);  1  p. 

ROHLAND,  P.  (1)  Uber  Plasticitat  der  Thone.  Z.  anorg.  Chem.  31,  158 
(1902);  3  pp.;  (2)  The  Means  for  Altering  the  Amount  of  Plasticity  of 
Clays.  Spreck.  39,  1371  (1906);  (3)  Explanation  of  Plasticity,  Bonding 
Power,  Shrinkage,  and  Adsorption  Properties  of  Clay.  (Due  to  col- 
loidal content);  (4)  Sprechsaal  47,  129  (1914);  1  p.;  (4)  The  Causes  of 
Plasticity  and  the  Allied  Properties  of  Clays  and  Kaolin.  Silikat 
Ztg.  2,  30  (1911);  3  pp. 

Rom,  A.  La  viscosita  e  1'elasticita  susseguente  nei  liquidi.  Cimento 
(3)3,5  (1878);  45  pp. 

RONCERAY,  P.  22,  35,  Flow  in  Capillary  Tubes.  Ann.  chim.  phys.  22, 
107  (1911);  19pp. 

ROSENCRANZ,  A.     Cp.  O.  E.  Meyer. 

ROSENOW,  M.  The  Plasticity  of  Clay.  Konigl.  Tech.  Hochsch.  Erlangen 
(1911);  Review  in  Tonind.  Ztg.  35,  1261  (1911);  1/4  p. 

ROSSANDER,  G.  Om  Gasers  Utstromning  Genom  Kapillarror  vid  Laga 
Tryck.  Oefvers.  Vet.  Akad.  Forhandl.,  Stockholm  (1900). 

ROSSEM,  V.    On  the  "Tackiness"  of  India  Rubber.     Kolloid-Z.  12,78(1913). 

VAN  ROSSEM,  A.  The  Viscosity  of  Crude  Rubber  Solutions.  Kolloidchem. 
Beihefte  10,  83  (1918);  46  pp. 

Rossi,  G.  284,  (1)  Sulla  viscosita  degli  idrosole  in  generate  e  sulla  funzione 
di  essa  negli  esseri  viventi.  Arch,  di  Fisiol.  3,  507  (1906);  25  p.;  (2) 
Sulla  viscosita  di  alcuni  colloidi  inorganici.  Arch,  di  Fisiol.  2,  246 
(1905);  (3)  La  viscosita  e  la  resistanza  elettrica  del  siero  di  sangue  a 
temperature  diverse  e  prossimo  a  quella  del  1'organismo.  Arch,  di  Fisiol. 
1,  500  (1904);  5  pp.;  (4;  La  viscosita  e  1'azione  denaturante  del  calore 
in  soluzioni  di  sieroalbumina.  Arch,  di  Fisiol.  2,  272  (1905);  (5) 
Sulla  temperatura  e  sul  tempo  di  coagulazione  delle  proteine  del  siero 
di  sangue  in  rapporto  con  la  viscosita  di  questo.  Arch,  di  Fisiol.  2,  599 
(1905).  Cp.  Fano. 

ROTHLIN,  E.  (1)  The  Technic  of  the  Estimation  of  the  Viscosity  of  Organic 
Colloids.  Biochem.  Z.  98,  34  (1919);  59  pp.;  (2)  Critical  Studies  of 
the  Rate  of  Flow  in  the  Determination  of  the  Viscosity  of  Blood  and  its 
Components.  Z.  klin.  Med.  89,  233  (1920);  41  pp. 

ROTHMUND,  V.  95,  Studien  iiber  die  kritische  Triibung.  Z.  physik.  Chem.. 
63,54  (1908);  29  pp. 

ROTINJANZ,  L.  Die  Zahigkeitsanderung  des  fliissigen  Schwefels.  Z.  physik. 
Chem.  62,  609  (1908);  11  pp. 

Roux,  J.  Stokes'  Law  and  the  Charge  of  the  Electron.  Compt.  rend. 
155,  1490  (1913);  4  pp. 

RUBINSTEIN,  H.  Untersuchungen  iiber  Beziehungen  zwischen  Schwer- 
schmelzbarkeit  und  Plastizitat  der  Tone.  Kommissionsverlag  der 
Tonindustrie  Ztg.  Berlin  (1920);  78  pp. 

RUCKES,  W.     51,  Untersuchung  iiber  den  Ausfluss  komprirnierter  Luft  aus 


412  INDEX 

Kapillaren  und  die  dabei  auftretenden  Turbulenzerscheinungen.     Ann. 

Phys.  25,  983  (1908);  39  p. 
RUDORF,  G.     (1)  Zur  Kenntniss  der  Leitfahigkeit  und  inneren  Reibung  von 

Losungen.     Z.  physik.  Chem.  43,  257   (1903);  47  pp.;  Diss.  Breslau 

(1903);  (2)  t)ber  die  innere  Reibung  von  Losungen.     Z.  Elektrochem. 

10,  473  (1904);  11  pp. 
RUDSKI,  P.     Note  on  the  Flow  of  Water  in  a  Straight  Pipe.     Phil.  Mag. 

(5)  36,  439  (1893);  2  pp. 
RUCKER.     69. 

SACC.     Sur  1'essai  des  gommes  employ6es  pour  6paissir  les  couleurs.     Bull. 

Soc.  Ind.  Mulhouse  28,  104. 
SACHANOV,    A.    &   RYACHOVSKII,   N.     Viscosity   of  Liquid    Mixtures.     Z. 

physik.  Chem.  86,  529  (1913);  J.  Russ.  Phys.  Chem.  Soc.  46,  78  (1914); 

10pp. 
SACHER,   J.     Determination  of  Viscosity  of  oils.     Farben  Ztg.    18,  2475 

(1913);  1  p. 
SACHS,  J.     t)ber  den  Einfluss  der  Dichtigkeit  auf  die  Viscositat  tropfbarcr 

Fliissigkeiten.     Diss.  Freiburg  (1883);  33  pp.     Cp.  Warburg  and  Sachs. 
SACKTJR,  O.     Das  elektrische  Leitvermogen  und  die  innere  Reibung  von 

Losungen  Caseins.     Z.  physik.  Chem.  41,  672  (1902);  9  pp. 
SAHLBORN,  N.     Koll.  Chem.  Beih.  2,  Heft  3  (1910). 
SALOMON,  L.     Description  of  the  Barbet   "Ixometre."     Revue  Gene"rale 

des  chemin  de  Fer. 
SAMELE,   E.     Viscosity  of  the  Blood  in  Glucosuria.     La.  clin.  med.  ital 

48,  162  (19--);  14pp. 
SAMMET,  C.  F.     Relative  Viscosity   of  Oils   at   Room   Temperature.     J.. 

Ind.  Eng.  Chem.  10,  632  (1918);  1  p. 
SANDERS,  H.     Motion  of  a  Viscous  Liquid  under  a  Rotating  Plate.     Ber. 

physik.  Ges.  14,  799  (1912);  6  pp. 

SAPH  &  SCHRODER.     Proc.  Am.  Soc.  Civ.  Eng.  61,  253  (1903). 
SAYBOLT.     324. 
SCARP  A,   O.     96,  100,    (1)   Determinazione  della  viscosita  del  fenolo  allo 

Btato  liquido.     Cim.   (5)  5,  117  (1903);  14  pp.;   (2)  Measurement  of 

the  Viscosity  of  Liquids  and  Lubricants.     Atti.  ist.  incorag.  Napoli 

(6)  7,  Gazz.  chim.  ital.  40,  II,  261  (1911);  25  pp.;  (3)  E  viscosita  o  la 
fluidita  una  proprieta  additiva?     Rend.   Soc.   Chim.  ital.   II  6,  363 
(1913);  8  pp.;  (4)  Sull'  influenza  dei  colloidi  sulla  viscosita  di  alcuni 
miscugli  binari  di  non  elettrolitti.     Do.   6,  370    (1913);    5    pp.;    (5) 
Transformazione  invertible  della  gomma  e  della  gelatina  dallo  stato  di 
emulsoide  a  quello  di  suspensoide  e  proprietA  di  tali  sistemi  p.     Do.  6, 
375  (1913);  5  pp.;  (6)  La  viscosite  des  solutions  d'eau  et  de  phenol. 
J.  chim.  phys.  2,  447  (1904). 

SCHAEFAR,  C.  &  FRANKENBERG.  The  Influence  of  Temperature  on  the 
Turbulent  Flow  of  Liquids.  Physik.  Z.  14,  89  (1913);  4  pp. 

SCHAEFER,  C.  (1)  t)ber  den  Einfluss  der  Temperatur  auf  die  Elasticitat 
der  Metalle.  Ann.  Physik.  6,  220  (1901);  14  pp.;  (2)  ttber  den  Einfluss 


INDEX  413 

der  Temperatur-  auf  die  Elastizitat  der  Eleinente.     Ann.  Physik.  9, 

665  (1902);  11  pp.;  Ann.  Physik.  9,  1124  (1902). 
SCHAEFFER  &  BuDENBERG.     Apparatus  for  the  Determination  of  Viscosity. 

Ger.  Pat.  234,941,  July  23  (1910). 
SCHALL,  C.     (1)  Uber  die  Reibung  von  Losungen  einiger  Ester  in  unter- 

kiihlten  Thymol.     Z.  physik.  Chem.  29,  423  (1899);  5  pp.;  (2)  Uber  die 

Zahigkeit  einiger  Losungen  welche  sich  aus  organischen  Substanzen 

zusammensetzen.     Physik.  Z.  3,  62  (1901);  (3)  Zahigkeit  von  unter- 

kuhlten  Losungen  in  Thymol.     Physik.  Z.  7,  645  (1906);  3  pp.;   (4) 

The  Change  in  Viscosity  due  to  Solution.     Z.  Elektrochem.  18,  225 

(1913). 
SCHALL,  C.  and  VAN  RUN,  W.     Uber  Reibung  von  Losungen  in  Glycerin. 

Z.  physik.  Chem.  23,  329  (1897);  20  pp. 
SCHAUFELBERGER,  W.     Bemerkungen  zu  der  Arbeit  des  Herrn  Quincke: 

Die  Klebrigkeit  isolirender  Fliissigkeiten  im  Constanten  electrischen 

Felde.     Wied.  Ann.  (3)  66,  635  (1898);  5  pp. 
SCHAUM,  K.     Uber  hylotrop-isomere  Korperformen.     Ann.  308,  18  (1899); 

22pp. 
SCHEITLIN,  W.     Vergleichende  Untersuchungen  liber  die  Blutviscositat  bei 

gesunden  und  kranken  Tieren.     Diss.  Zurich  (1909);  83  pp. 
SCHENCK,    R.     209,    (1)   Untersuchungen  liber  die  krystallinischen  Fliis- 

sigkeiten.     Innere  Reibung.  Z.  physik.  Chem.  27,  167  (1898);  5  pp.; 

(2)     Kristallinische    Fliissigkeiten    und    fllissige    Kristalle.     Leipzig. 

W.  Engelmann  (1905);  159  pp. 
SCHEUER,  O.     Physikochemische  Studien  an  binaren  Gemischen  mit  einer 

optisch-aktiven    Komponente.     Z.    physik.    Chem.    72,    546    (1910); 

21pp. 
SCHEURER,  F.     Sur  le  viscosimetre  de  M.  le  professeur  G.  Lunge.     Bull. 

Soc.  Ind.  Mulhouse  66,  57  (1896);  7  pp.     Cp.  Lunge. 
SCHIDROWITZ,  P.  &  GOLDSBROXJGH,  A.  H.     (1)  Die  Viskositat  von  Gummi 

und  Gummilosungen  mit  besonderer  Berticksichtgung  der  Beziehung 

zu  der  Starke  oder  den  "Nerv"  von  Rohgummi.     Kolloid.-Z.  4,  226 

(1909);  6  pp.;  J.  Soc.  Chem.  Ind.  28,  3  (1909);  (2)  Viscosity  of  Solutions 

of  Rubber.     Caoutchouc  &  Guttapercha  9,  6220,  Kolloid.-Z.   13,  46 

(1913). 
SCHIERLOCH.     Uber  dem  Koefficienten  der  inneren  Reibung  von  reinem 

Argon  und  reinem  Helium.     Diss.  Halle  (1908) ;  30  pp. 
SCHILLER,  L.     Determination  of  Absolute  Viscosity  by  the  Glimbel  Viscom- 

eter.     Z.  techn.  Physik.  1,  72  (1920);  3  pp.;  do.,  2,  50  (1921);  2  pp. 
SCHITERG,  J.     The  Importance  of  Viscosity  Measurements  for  the  Knowl- 
edge   of   Organic  Colloids.     Internat.  Z.  physik.  chem.   Biol.   1,  260 

(1914). 
SCHLAMP,  A.     Zur  Dissociationtheorie  der   Losungen.     Z.    physik.    chem. 

14,  284  (1894);  14  pp.     (Viscosity  1  p.). 
SCHLIE.     179,  Untersuchungen  liber  die  Bewegung  von  Fliissigkeiten  in 

Capillarrohren.     Inaug.   Diss.   Rostock    (1869). 
SCHLTJMBERGER,  I.     Sur  la  notice  de  M.  le  Dr.  Sacc,  au  sujet  de  1'essai  des 


414  INDEX 

gommes  employees  pour  epaissir  les  couleurs.     Bull.    Soc.   Ind.   Mul- 

house  26,  114,  2  pp.     Cp.  Sacc.  Lunge,  etc. 
SCHMIDT,  H.     Capillary  Rise  in  Filter  Paper.     Kolloid-Z.  26,  152  (1920); 

8pp. 
SCHMIDT,  M.  and  JONES,  H.     Conductivity  and  Viscosity  in  Mixed  Solvents 

Containing  Glycerol.     Am.  Chem.  J.  42,  37  (1909) ;  58  pp.     Cp.  Jones. 
SCHMIDT,    P.     Uber   die  innere   Reibung   fester  Korper.     Wied.   Ann.    2, 

46,  241  (1877);  19  pp. 
SCHMIDT,   T.     6,   Bestimmung  der  Reibung  von   Flussigkeiten  nach  der 

Methode  von  Maxwell.     Diss.  Breslau   (1881);  Wied.  Ann.  16,  633 

(1882);  28  pp. 
SCHMITT,  K.     The  Viscosity  of  Certain  Gases  and  Gaseous  Mixtures  at 

Different  Temperatures.     Ann.  Physik.  30,  393  (1909);  17  pp.;  Ann. 

Physik.  30,  393  (1909);  18  pp.;  Diss.  Halle  (1909). 
SCHNEEBELI,  M.     La  valeur  absolue  du  coefficient  de  frottement  de  1'air. 

Arch.  sci.  phys.  nat.  (3)  14,  197  (1885) ;  6  pp. 
SCHNEIDER.     Diss.  Rostock. 
SCHO'TTNER,  F.     6,  179,  (1)  tTber  die  innere  Reibung  im  Glycerin.     Wien. 

Sitzungsber.  (2A;  77,  682  (1878);  17  pp.;  (2)  Uber  die  Ermittlung  des 

Coefficienten  der  inneren  Reibung  in  zahen  Flussigkeiten,  durch  Fall- 

versuche.     Wien.  Sitzungsber.  (2A)  79,  477  (1879);  14  pp. 
SCHROEDER.     Afhandlingen    ar    publicerad    i  en  rysk   tidskrift.     Mining 

Journal  2  (1888);  Diss.  Strassburg  (1886);  27  pp. 
VON  SCHROEDER,  P.     213,  tlber  Erstarrungs-  und  Quellungserscheinungen 

von  Gelatine.     Z.  physik.  Chem.  45,  75  (1903);  43  pp. 
SCHROEDER,  T.     Experimentaluntersuchung  iiber  den  Einfluss  der  Tem- 

peratur  auf  die  elastische  Nachwirkung.     Wied.  Ann.  28,  369  (1876); 

25pp. 
SCHRYVER,    S.    B.,   RAMSDEN,    W.,    SCHIDROWITZ,    ET  AL.     Discussion   of 

Colloids  and  their  Viscosity.     Trans.  Faraday   Soc.  9,  93  (1913);  14 

pp.;  Z.  Chem.  Ind.  Kolloide  12,  253  (1913) ;11  pp. 

SCHUTT,  K.     Uber  Zahigkeit  und  Festigkeit  in  der  Oberflache  von  Flussig- 
keiten und  uberfliissig  Lamellen.     Ann.  Physik.  (4)  13,  712  (1904);  35 

pp. 
SCHULE,   W.     Air  Resistance  in   the  Light   of  Recent  Experiments.     Z. 

Vereines  duetsch.  Ing.  54,  12,  54,  94  (1910);  18  pp. 
SCHULZ.     Bin  schnell  Viscosimeter.     Chem.  Ztg.  32,  891  (1908). 
SCHUKOWA-FLORENSOWA,  M.     Die  Viscositat  des  Blutes  der  Kinder  bes- 

onders  der  Mundatmer.     Diss.  Bern.  (1910);  21  pp. 
SCHULTZE,  H.     79,  (1)  Die  innere  Reibung  von  Argon  und  ihre  Aenderung 

mit  der  Temperatur.     Ann.  Physik.  (4)  6,  140  (1901);  26  pp.;  Diss. 

Halle  (1901);  58  pp.;  (2)  Uber  die  innere  Reibung  von  Helium  und  ihre 

Anderung  mit  der  Temperatur.     Ann.  Physik.  (4)  6,  302  (1901);  13pp. 
SCHUMANN,  O.     246,  tlber  die  Reibungsconstante  von  Gasen  und  Dampfen 

und  ihre  Abhangigkeit  von  der  Temperatur.     Wied.   Ann.   23,  353 

(1884);  51  pp. 
SCHUYTEN,  M.     Viscositatsbestimmungen  von  wasserig  Antipyrinlosungen. 

Chem.  Ztg.  30,  18  (1906);  1  p. 


INDEX  415 

SCHWARTZ,  H.     (1)  Viscosity  and  its  Importance  for  the  Chemistry  of  Cellu- 
loid in  Theory  and  Practice.     Z.  Chem.  Ind.  Kolloid.  12,  32  (1913); 

10  pp.;  J.  Soc.  Chem.  Ind.  32,  191;  Moniteur  Scientifique,  49  (1915); 

(2)    Viscosity   of   Nitrocellulose   Solutions.     Z.    Chem.   Ind.    Kolloide 

12,  32  (1913);  (3)  Viscosity  and  Its  Meaning  in  the  Chemistry  of  Cellu- 
loid.    Koll.  Z.  12,  32  (191-);  10  pp.;  (4)  Celluloid:  The  Necessity  of 

Colloid-Chemical  Views  in  this  Industry.     Kolloidchem.  Beihefte  6, 

90  (1914);  36  pp. 
SCHWEDHELM,  H.     (1)  Calculating  the  Viscosity  of  Mixtures  of  Two  Oils  of 

Different  Viscosities  from  these  Viscosities.     Chem.  Ztg.  44,  638  (1920) ; 

(2)  Die  Zahigkeit  von  Olen  und  anderen  Fliissigkeiten  als  Funktion 

der  Temperatur.     Chem.  Ztg.  45,  41  (1921);  1  p. 
SCHWEDOFF,  T.     Recherches  expe"rimentales  sur  la  cohesion  des  liquides. 

II.  Viscosite"  des  liquides.     J.  de  phys.  (2)  9,  34  (1890);  13  pp.     Con- 

gres  de  Physique  1,  478  (1900). 
SCHWEIDLER,    E.    VON.     197,    tlber  die  innere  Reibung    und  Elektrische 

Leitungsfahigkeit   von  Quecksilber  und  einigen  Amalgamen.     Wien. 

Sitzungsber.  (2A)  104,  273  (1895);  8  pp. 
SEARLE,  A.  B.     Clays  and  Clay  Products.     Dept.  Sci.  Ind.  Research,  Brit. 

Assoc.  Adv.  Sci.  Third  Report  on  Colloid  Chem.  (1920);  42  pp. 
SEARLE,  G.  F.  C.     (1)  Simple  Viscometer  for  Very  Viscous  Liquids.     Proc. 

Camb.  Phil.  Soc.  16,  600  (1913);  6  pp.;  (2)  A  Method  of  Determining 

Viscosity  of  Air.     Proc.  Camb.  Phil.  Soc.  17,  II,  153  (1913);  39  pp. 
SEATON,  M.,  PROBECK,  E.,  and  SAWYER,  G.     Viscosity  of  Varnishes.     J. 

Ind.  Eng.  Chem.  9,  35  (1917);  5  pp. 
SEELIS,  K.     Measurement  of  the  Brownian  Molecular  Motion  as  a  function 

of  Viscosity.     Z.  physik.  Chem.  86,  682  (1913);  57  pp. 
SEGEL,  M.     235,  Uber  eine  Methode  zur  Bestimmung  der  inneren  Reibung 

fester  Korper.     Physik.  Z.  4,  493  (1903);  2  pp. 
SELLARIO,  A.     Viscosity  of  Insulating  Liquids  in  the  Electric  Field.     Nuovo 

cimento  11,  395  (1916);  16  pp. 
SENTER,   G.     Viscosity  and  Association  in   Binary   Mixtures  of  Liquids. 

Proc.  Chem.  Soc.  25,  292  (1910);  3  pp. 
SHAEFER,  C.  &  FRANKENBERG,  G.     The  Influence  of  Temperature  upon 

Turbulent  Currents.     Physik.  Z.  14,  89  (1913);  5  pp. 
SHARPLES,    P.     Relation   Between   the   Melting   Point   and   Viscosity    of 

Refined  Tars.     J.  Ind.  Eng.  Chem.  6,  285  (1914);  1  p.;  Am.  Gas  Light 

J.  100,  247  (1914);  1  p. 

SHAW,  W.     Proc.  Camb.  Phil.  Soc.  7,  21  (1892). 
SHEPPARD,  S.  E.     329,  Measurement  of  the  Absolute  Viscosity  of  Very 

Viscous  Media.     J.  Franklin  Inst.  182,  697  (1916);  2  pp.;  J.  Ind.  Eng. 

Chem.  9,  523  (1917);  4  pp. 
SHEPPARD,  S.  E.  and  SWEET,  S.  S.     The  elastic  properties  of  gelatin  jellies. 

J.  Amer.  Chem.  Soc.  43,  539  (1921);  8  pp. 
SHORY,  U.  S.     The  Effect  of  the  Method  of  Preparation  on  the  Viscosity 

of  a  Casting  Slip.     J.  Am.  Ceram.  Soc.  3,  286  (1920);  10  pp. 
SHUKOWSKI,  N.  E.     (1)  Hydrodynamische  Theorie  der  Reibung  gut  gefett- 

eter  fester  Korper.     J.  d.  Russ.  Phys.  Chem.  Gesell  18,  209  (1886); 


416  INDEX 

(2)  Apparat  zur  Bestimmung  der  Koefficienten  der  Viskositat  von 
Fliissigkeiten.  Arbeiten  d.  Phys.  Sekt.  d.  Moskauer  Ges.  von  Freunden 
der  Naturf.  (4)  1,  25  (1891). 

SHULENBERGER,  F.  W.  (1)  Viscometers.  Paint,  Oil  and  Chem.  Rev. 
72,  #3,  10  (1921);  3  pp.;  (2)  The  Improved  MacMichael  Viscometer. 
Do.,  72,  #7,  10  (1921);  3  pp.;  (3)  Bingham  and  Green  Plastometer. 
Do.,  72,  #9  (1921);  2  pp. 

SIDGWICK,  N.  PICKFORD,  and  WILSDON,  B.  The  Solubility  of  Aniline  in 
Aqueous  Solutions  of  its  Hydrochloride.  J.  Chem.  Soc.  99,  1131 
(1911);  2  pp. 

SIDGWICK,  N.  V.  and  TIZARD,  H.  T.  The  Color  and  lonization  of  Cupric 
Salts.  J.  Chem.  Soc.  Lond.  97,  964  (1910);  16  pp. 

SIDGWICK,  N.  V.  &  WILSDON,  B.  H.  Conductivity  and  Viscosity  of  Aque- 
ous Solutions  of  Aniline  Hydrochloride  at  25°.  J.  Chem.  Soc.  99, 
1118  (1911);  5  pp. 

SIMEON,  F.  The  Viscosity  of  Calcium  Chloride  Solutions.  Phil.  Mag. 
(6)27,95  (1913);  5  pp. 

SIMON,  D.  (1)  Richerche  sulla  coagulazione  delle  albumine.  I.  Varia- 
zioni  fisico-chimiche  del  siero  per  aggiunta  di  alcool.  Arch  di  Fisiol. 
4,  594  (1907);  Do.,  6,394  (1908);  Do.,  6,402  (1908);  Do.,  5,  470  (1908); 
Do.,  6,  477  (1908);  Do.,  6,  479  (1908). 

SIMONIS,  M.  232,  233,  234,  (1)  Zur  Zahigkeitsmessung  von  Tonbreien. 
Sprech.  38,  597  (1905);  6  pp.; -(2)  Zettlitzer  Erde  und  Alkali.  Zur 
Theorie  des  Giessverfahrens.  Sprech.  38,  881  (1905);  3  pp.;  (3) 
Physikalische  Definition  der  Tone  und  Magerungsmittel.  Sprech. 
38,  1625  (1905);  2  pp.;  (4)  Weitere  Beitrage  zum  Verhalten  von  Tonen 
und  Magermitteln  gegen  Elektrolyte.  Sprech.  39,  1167  (1906);  3  pp.; 
(5)  Do.,  Sprech.  39,  1184  (1906). 

SINGER,  L.  (1)  The  Lubrication  of  Machine  Parts.  Petroleum  7,  1307 
(1913);  13  pp.;  (2)  Untersuchungen  der  Schmiermittel,  Osterr.  Chem. 
Ztg.  (1906). 

SKAUPY,  F.  The  Specific  Heat  of  Liquid  Mercury.  The  Heat  Content  of 
Liquids,  Especially  Metals,  at  the  Melting  Point,  and  Its  Relation  to 
Specific  Heat,  Electrical  Conductivity,  and  Internal  Friction.  Ber. 
deut.  physik.  Ges.  18,  302  (1916);  5  pp. 

SLATOWRATSKY,  N.  &  TAMMANN,  G.  Erweichen  Kristalle  in  der  Nahe 
ihres  Schmelzpunktes.  Z.  physik.  Chem.  63,  341  (1905);  8  pp. 

SLOTTE,  K.  6,  128,  130,  142,  178,  179,  (1)  tlber  die  innere  Reibung  der 
Losungen  einiger  Chromate.  Wied.  Ann.  14,  13  (1881);  (2)  Uber  die 
innere  Reibung  einiger  Losungen  und  die  Reibungsconstante  des 
Wassers  bei  verschiedenen  Temperaturen.  Wied.  Ann.  20,  257  (1883); 
11  pp.;  (3)  Om  den  inre  friktionen  hos  vatskor.  Oefvers.  af  Finska 
Vetensk.  Soc.  Forhandl.  32,  116  (1890);  33  pp.;  (4)  tJber  Reibungs- 
konstanten  und  andere  Konstanten  der  Fliissigkeiten.  Oefvers.  af. 
Finska  Vetensk.  Soc.  Forhandl.  37,  11  (1895);  Cp.  Wied.  Beibl.  16, 
182;  (5)  Uber  die  Elastizitat  der  Metalle.  Acta  Soc.  Fenn.  36,  14 
(1908). 


INDEX  417 

SMITH,  A.  and  HOLMES,  W.  and  HALL,  E.     Uber  den  amorphen  Schwefel. 

Z.  physik.  Chem.  52,  606  (1905);  24  pp. 
SMITH,  A.  and  MENZIES,  A.     The  Electrical  Conductivity  and  Viscosity  of 

Concentrated    Solutions    of    Orthophosphoric    Acid.     J.    Am.    Chem. 

Soc.  31,  1191  (1909);  3  pp. 
VON  SMOLUCHOWSKI.     96,  188,  (1)  Viscosity  of  Colloidal  Solutions.     Koll- 

oid-Z.  18,  190  (1916);  (2)  Uber  die  innere  Reibung  in  nicht  wasserigen 

Losungen.     Wien.  Sitzungsber  (2A)  1136  (1893);  5  pp.;  (3)  Theory  of 

the  Liquid  State  from  the  Standpoint  of  the  Phenomena  of  Viscosity. 

Kosmos  35  (Radziszevski  Festband)  543  (1910);  7  pp. 
SNYDER,  C.  D.  &  TODD,  M.  H.     The  Viscosity  of  Body  Fluids  at  Various 

Temperatures  within  Physiological  Limits.     Am.  J.  Physiol.  28,  161 

(1911);  6  pp. 
SNYDER,  L.  C.     Theories  Concerning  the  Plasticity  of  Clays.     Mining  Sci. 

64,  106  (1911);  3pp. 
SOMMERFELD,  A.     264,  268,  (1)  Zur  hydrodynamischen  Theorie  der  Schmier- 

mittelreibung.     Z.  f.  Math.  u.  Phys.  50,  97  (1904);  58  pp.;  (2)  Die 

naturwissenschaftlichen  Ergebnisse  und  die  Ziele  der  modernen  tech- 

nischen  Mechanik.     Physik.  Z.  4,  779  (1903);  3  pp. 
SONDHAUS,    C.     tJber   die   Form  vori   aus   runden   Oeffnungen   tretenden 

Luftstromen.     Pogg.  Ann.  85,  58  (1852);  6  pp. 
SORKAU,   W.     (1)   Turbulence   Viscosity  of  Water.     Physik.   Z.    14,   759 

(1913);  7  pp.;  Do.,  14,  826  (1913);  5  pp.;  (2)  Influence  of  Temperature 

Specific  Gravity  and  Chemical  Nature  on  the  Turbulence  Viscosity. 

Physik.  Z.  13,  805  (1913);  16  pp.;   (3)  Relation  between  Molecular 

Weight  and  Turbulence  Velocity  Constant.     Physik.  Z.  14,  147  (1913); 

6  pp.;  Physik.  Z.  15,  582  (1913);  5  pp.;  (4)  Experimental  Investigation 

of  the  Viscosity  of  Some  Organic  Liquids  in  the  State  of  Turbulent 

Flow.     Physik.  Z.  12,  582  (1911);  14  pp. 
SOUTH  COMBE,  J.  E.     Lubrication.     Engineering  109,  184  (1920);  1  p.     Cp. 

Wells. 
SOUTHERN  PACIFIC  R.  R.  Co.  258,  A  Long  Rifled-Pipe  Line  for  Pumping 

Heavy  Oils.     Engineering  Record  57,  676  (1908). 
SPERANSKII  &  KARAVEYER.     Viscosity  of  Certain  Petroleum  Distillates  at 

Low  Temperatures.     Nephtanoje  Djelo,  14  (1912);  Chem.  Ztg.  Rep. 

36,  482. 
SPRIGGS,  E.   I.     286,  Eine  neue  Methode  zur  Bestimmung  der   Pepsin- 

wirkung.     Hoppe-Seyler's  Zeitsch.  f.  Physiol.  Chem.  35,  465  (1902); 

29  pp. 
SPRING,    W.     (1)    t)ber   das   Vorkommen   gewisser   fur   den   Fliissigkeits 

oder  Gaszustand  charakteristischen  Eigenschaften  bei  festen  Metallen. 

Z.  physik.  Chem.  15,  65  (1895);  14  pp.;  (2)  Eigenschaften  der  festen 

Korper  unter  Druck,  Diffusion  der  festen  Material,  innere  Bewegungen 

der   festen    Materie.     Physik.    Z.    1,   487    (1900);    Cp.    International 

Physical  Congress  at  Paris,  Report  1,  402  (1900);  1  p. 
SPROXTON,  F.     Cellulose  Esters.     Dept.  Sci.  Ind.  Research,  Brit.  Assoc. 

Adv.  Sci.  Third  Report  on  Colloid  Chem.  (1920);  10  pp. 


418  INDEX 

SPRUNG,  A.  6,  178,  179,  Experimentelle  Untersuchungen  iibcr  die  Fliis- 
sigkeitsreibung  bei  Salzlosungen.  Pogg.  Ann.  169,  1  (1876);  35  pp. 

STABLES,  W.  and  WILSON,  A.  6,  254,  Experiments  on  the  Viscosity  of  a 
Solution  of  Saponine.  PhiL  Mag.  (5)  16,  406  (1883);  9  pp. 

STALEY,  H.  F.  The  Viscosity  of  Borate  Glasses.  Orig.  Com.  8th  Inter. 
Congr.  Appl.  Chem.  6,  127  (1913);  11  pp. 

STANCE,  M.  Apparat  zur  Bestimmung  der  Zahfliissigkeit.  Chem.  Ztg. 
30,643  (1906);  1  p. 

STANKEWITSCH.     Warschauer  Universitats.     Nachrichten  (1887)  (Russian). 

STANTON,  T.  (1)  Some  Characteristics  of  the  Flow  of  Water  in  Channels 
of  Varying  Cross-Section.  Engineering  74,  664  (1902);  2  pp.;  (2) 
The  Law  of  Comparison  for  Surface  Friction  and  Eddy-Making  Resis- 
tance in  Fluids.  Nat.  Phys.  Lab.  Coll.  Res.  9,  1  (1913);  8  pp.;  (3) 
The  Mechanical  Viscosity  of  Fluids.  Proc.  Roy.  Soc.  London  (A) 
85,  366  (1911);  11  pp.;  (4)  Determination  of  the  Absolute  Viscosities 
of  Liquids  at  High  Pressures.  Engineering  108,  520  (1919);  2  pp.; 
Cp.  Do.,  108,  758  (1919);  3  pp. 

STANTON,  T.  &  PARNELL,  J.  Similarity  of  Motion  in  Relation  to  the  Surface 
Friction  of  Fluids.  Phil.  Trans.  214  (A),  199  (1914);  25  pp.  Nat. 
Phys.  Lab.  Coll.  Researches  11,  295  (1914). 

STEELE,  B.,  MC!NTOSH,  D.,  and  ARCHIBALD.  183,  Halogen  hydrides  as 
Conducting  Solvents.  I.  Vapor  Pressures,  Densities,  Surface  Energies, 
and  Viscosities  of  the  Pure  Solvents.  Phil.  Trans.  London  (A)  206, 
99  (1906);  70pp. 

STEPHAN,  C.  Beitrage  zu  den  Beziehungen  zwischen  Fluiditat  und  galvan- 
ischen  Leitungsvermogen.  Diss.  Breslau  (1882);  Wied.  Ann.  17, 
673  (1882);  29  pp. 

KTEPHAN,  J.  6,  14,  (1)  tlber  die  Bewegung  flussiger  Korper.  Wien.  Sitz- 
ungsber  (2A)  46,  8  (1862);  24  pp.;  (2)  Do.,  Wien.  Sitzungsber.  46, 
495  (1862);  26pp. 

STEUDEL,  V.  tJber  Transpiration  von  Dampfen.  III.  Wied.  Ann.  16, 
369  (1882);  26  pp.  Cp.  Meyer. 

STEVENS,  H.  P.  (1)  Fractional  Separation  of  Rubber  by  Solution  in  Benzene 
and  the  Viscosity  of  the  Fractions  in  Benzene  solution.  India  Rubber 
J.  46,  345  (1913);  Cp.  Beadle;  (2)  The  Viscosity  of  Latex  and  Its  Bearing 
on  the  Rate  of  Cure.  Bull.  Rubber  Growers'  Assoc.  2,  214  (1920) ;  2  pp. 

STEVENS,  J.     Viscosity  of  Oils.     J.  Soc.  Chem.  Ind.  33,  109  (1913) ;  2  pp. 

STEWART,  B.  and  TAIT,  P.  (1)  On  the  Heating  of  a  Disk  by  Rapid  Rota- 
tion in  Vacua.  Proc.  Roy.  Soc.  London  14,  339  (1865);  5  pp.;  (2) 
Do.,  Proc.  Roy.  Soc.  London  15,  290  (1866);  10  pp. 

STEWART,  G.  W.  A  Noteworthy  Interrelation  of  Illuminating  Power, 
Density  and  Viscosity  of  Certain  Kerosene  Oils.  Phys.  Rev.  21,  513 
(1912);  11  pp. 

STEWART,  J.  The  Plasticity  of  Clay.  Orig.  Com.  8th  Interm.  Congr. 
appl.  Chem.  15,  265  (1913);  6  pp.;  J.  Ind.  Eng.  Chem.  6,  421  (1913); 
2pp. 

STOCK.     Bull,  de  1'Acad.  des  Sci.  de  Cracovie  1  (A),  ,18  (1911). 


INDEX  419 

STOCKS,  H.  B.  Colloid  Chemistry  of  Starch,  Gums,  Hemicelluloses, 
Albumin,  Casein,  Gluten  and  Gelatine.  Dept.  Sci.  Ind.  Research, 
Brit.  Assoc.  Adv.  Sci.  First  Report  on  Colloid  Chem.  (1917);  33  pp. 

STOEL.  130,  Measurements  on  the  Influence  of  Temperature  on  the 
Viscosity  of  Fluids  between  the  Boiling-Point  and  the  Critical  State. 
Diss.  Leiden  (1891);  Physikalische  Revue  (Graetz)  1,  513  (1892); 
Communications  from  the  Laboratory  of  Physics,  Leiden  2,  (1891). 

STOKES,  G.  2,  6,  14,  188,  194,  264,  (1)  On  Some  Cases  of  Fluid  Motion. 
Phil.  Trans.  Camb.  II  8,  105  (1849);  (2)  Supplement  to  a  Memoir  on 
Some  Cases  of  Fluid  Motion.  Phil.  Trans.  Camb.  II  8,  287  (1849); 
(3)  On  the  Theories  of  the  Internal  Friction  of  Fluids  in  Motion. 
Phil.  Trans.  Camb.  II  8,  409  (1849);  (4)  On  the  Effect  of  the  Internal 
Friction  of  Fluids  on  the  Motion  of  Pendulums.  Phil.  Trans.  Camb. 
II  9,  8  (1851);  99  pp.;  (5)  Do.  Phil.  Mag.  (4)  1,  337  (1851);  (6)  Note 
on  the  Reduction  of  Mr.  Crookes'  Experiments  on  the  Decrement  of 
the  Arc  of  Vibration  of  a  Mica  Plate  Oscillating  within  a  Bulb  Contain- 
ing a  More  or  Less  Rarefied  Gas.  Phil.  Trans.  London  B  172,  435 
(1881);  Cp.  Proc.  Roy.  Soc.  London  31,  458  (1881);  (7)  Note  on  a  Paper 
by  H.  Tomlinson,  just  preceding.  Phil.  Trans.  B  177,  786  (1886); 
3  pp.;  Cp.  Mathematical  and  Physical  Papers,  Cambridge  Univ.  Press 
(1883). 

STORMER.     328,  Viscometer.     Drugs,  Oils  and  Paints  27  (1911). 

STOVER,  E.  C.  Die  Fortpflanzung  von  Bakterien  als  Ursache  der  Plas- 
tizitat  der  Tone.  Deutsche  Topfer-und  Ziegler  Ztg.  No.  14  (1903). 

STRASBTJRGER,  C.  F.  Apparatus  for  Testing  the  Viscosity  of  Liquids. 
U.  S.  Pat.  989,  822,  April  18  (1909). 

STREINTZ,  H.  237,  (1)  Uber  die  Dampfung  der  Torsionsschwingungen  von 
Drahten.  Pogg.  Ann.  163,  387  (1874);  25  pp.;  (2)  Wien.  Sitzungsber. 
(2A)  69,  337  (1874);  (3)  Wien.  Sitzungsber.  (2A)  80,  397  (1880). 

STREVENS,  J.     Viscosity  of  Oils.     J.  Soc.  Chem.  Ind.  33,  109  (1913);  2  pp. 

STRIBECK,  R.  Mitteilungen  uber  Forschungsarbeiten.  Heft.  7  (1903); 
Springer;  Z.  d.  Ver.  d.  Ingen  46,  1341  (1902). 

STROM,  L.  Separating  Mixed  Liquids  of  Different  Viscosities.  U.  S.  Pat. 
968,206,  Aug.  23  (1909). 

STULL,  R.  T.  Fluxing  Power  of  Mica  in  Ceramic  Bodies.  Trans.  Am.  Cer. 
Soc.  4,  255  (1902);  16pp. 

SUTHERLAND,  W.  196,  246,  247,  248,  (1)  The  Viscosity  of  Gases  and  Mole- 
cular Force.  Phil.  Mag.  (5)  507  (1893);  25  pp.;  (2)Ionization  in 
Solutions  and  Two  New  Types  of  Viscosity.  Phil.  Mag.  (6)  14,  1 
(1907);  35  pp.;  (3)  Phil.  Mag.  (6)  9,  781  (1905);  Austr.  Assoc.  10th 
Meet.,  Dunedin  (1904);  p.  117  (1905). 

SWEDBERG,  T.  Velocity  of  Diffusion  and  Size  of  the  Particles  in  Disperse 
Systems.  Archiv.  Kern.  Min.  Geol.  4,  #12,  1  (1914);  6  pp.;  J.  Chem. 
Soc.  102,  II,  142  (1913). 

SWEDBERG,  T.  &  SWEDBERG,  A.  Diffusionsgeschwindigkeit  und  relative 
Grosse  geloster  Molecule.  Z.  physik.  Chem.  76,  145  (1913). 


420  INDEX 

TAIT,  P.  On  the  Foundations  of  the  Kinetic  Theory  of  Gases.  Trans. 
Roy.  Soc.  Edinburgh  33,  259  (1887);  27  pp. 

TAMMANN,  G.  6,  188,  (1)  Velocity  of  Solidification  of  Liquids.  Z. 
physik.  Chem.  23,  326  (1897);  3  pp.;  (2)  Uber  die  Viscositat  unter- 
kuhlter  Fliissigkeiten.  Z.  physik.  Chem.  28,  17  (1898);  15  pp.;  Cp. 
Werigen,  Lewkojeff,  and  Tammann;  (3)  tJber  die  Ausflussgeschwindig- 
keit  krystallisirter  Stoffe.  Ann.  Physik.  7,  198  (1902);  27  pp.;  (4) 
Kristallisieren  und  Schmelzen,  p.  158;  (5)  Sprechsaal  (1904);  p.  35;  (6) 
The  Velocity  of  Crystallization.  Z.  Physik.  Chem.  81,  171  (1913); 
6pp. 

TANZLER,  P.  Koefficienten  der  inneren  Reibung  fur  Gemische  zwischen 
Argon  und  Helium.  Diss.  Halle  (1906);  Verh.  D.  physik.  Gesell.  8, 
222  (1906);  13  pp. 

TAPPEINER.  Uber  die  Wirkung  der  Mucilaginosa.  Arch,  internat.  d. 
Pharmacodynamie  (1902). 

TAYLOR,  F.  W.  On  the  Art  of  Cutting  Metals.  Am.  Soc.  Mech.  Eng., 
N.  Y.,  248  pp. 

TAYLOR,  W.  Note  on  the  Standard  of  Relative  Viscosity  and  on  Negative 
Viscosity.  Proc.  Roy.  Soc.  Edinburgh  25,  227  (1904);  3  pp. 

TAYLOR,  W.  and  MOORE,  T.  On  the  Negative  Viscosity  of  Aqueous 
Solutions.  Proc.  Roy.  Soc.  Edinburgh  28,  461  (1907);  11  pp. 

TAYLOR,  W.  and  RANKEN,  C.  (1)  The  Viscosity  of  Aqueous  Solutions  of 
Chlorides,  Bromides,  and  Iodides.  Proc.  Roy.  Soc.  Edinburgh  25, 
231  (1904);  10  pp.;  Cp.  Ranken  and  Taylor;  (2)  Note  on  the  Standard 
of  Relative  Viscosity  and  on  "Negative"  Viscosity.  Proc.  Roy.  Soc. 
London  25,  227  (1904). 

TEMPLE,  J.  W.     297,  298,  Thesis  Lafayette  College  (1921). 

THIESSEN,  M.  Reibung  vom  Gasgemischen.  Verh.  D.  physik.  Gesell. 
8,236  (1906);  2  pp. 

THOLE,  F.  B.  112,  (1)  Viscosity  and  Association.  I.  The  Association  of 
the  Phenols.  Proc.  Chem.  Soc.  26,  328  (1911);  J.  Chem.  Soc.  97, 
2596  (1911);  11  pp.;  II.  The  Viscosity  of  Geometrical  Isomerides. 
Proc.  Chem.  Soc.  28,  51  (1912);  J.  Chem.  Soc.  101,  552  (1912);  6  pp.; 
III.  Existence  of  Racemic  Compounds  in  the  Liquid  State.  J.  Chem. 
Soc.  103, 19  (1913);  8  pp.;  Proc.  Chem.  Soc.  28,  286  (1913);  IV.  Viscosity 
of  Aromatic  Amines.  J.  Chem.  Soc.  103,  317  (1913);  6  pp.;  Proc. 
Chem.  Soc.  29,  32  (1913);  (2)  Note  of  Anomalous  Viscosity  of  Nitro- 
benzene. Proc.  Chem.  Soc.  25,  198  (1910);  (3)  The  Viscosity  of 
Isodynamic  and  Moto-isomers.  Z.  Physik.  Chem.  74,  683  (1910);  3 
pp.  Cp.  Dunstan. 

THOLE,  F.  B.,  MUSSEL,  A.  G.  &  DUNSTAN,  A.  E.  Viscosity  Maxima  and 
Their  Interpretation.  J.  Chem.  Soc.  103,  1108  (1913);  1  p.;  Proc. 
Chem.  Soc.  29,  174  (1913). 

THOMLINSON,  H.  Viscosity  of  Solids,  Damping  of  Vibrations,  Due  to 
Viscosity  under  Various  Circumstances.  Phil.  Trans.  London  177, 
801  (1887);  36  pp. 

THOMSEN,  E.  The  Viscosity  of  Gas  Mixtures.  Ann.  Physik.  36,  815 
(1911);  18pp. 


INDEX  421 

THOMSON,  J.  6,  262,  (1)  On  the  Charge  of  Electricity  Carried  by  Ions 
produced  by  Rontgen  Rays.  Physic.  Rev.  8,  141  (1899);  Phil.  Mag. 
(5)  46,  528  (1898);  17  pp.;  (2)  Do.  Phil.  Mag.  (6)  5,  346  (1903);  10 
pp. 

THORPE,  E.  Viscosity  of  Pure  Liquids.  Science  Progress  12,  583  (1918); 
8pp. 

THORPE,  T.  and  RODGER,  J.  2,  6,  63,  64,  70,  71,  72,  82,  90,  107  et  seq.  113, 
122,  127,  130,  142,  160,  166,  169, 175,  (1)  On  the  Relations  between  the 
Viscosity  of  Liquids  and  their  Chemical  Nature.  Phil.  Trans.  London 
A  185,  397  (1894);  314pp.;  Cp.  Proc.  Roy.  Soc.  London  55, 148  (1894); 
and  Z.  physik.  Chem.  14,  361  (1894);  13  pp.;  (2)  Do.,  Part  II.  Phil. 
Trans.  London  A  189,  71  (1897);  36  pp.;  Cp.  Proc.  Roy.  Soc.  London 
60,  152  (1896);  Chem.  News  75,  152  (1897);  (3)  On  Some  Recent 
Results  of  Physico-Chemical  Inquiry.  Proc.  Roy.  Inst.  Gt.  Brit. 
15,  641  (1898);  19  pp.;  (4)  The  Viscosity  of  Mixtures  of  Miscible 
Liquids.  J.  Chem.  Soc.  71,  360  (1897). 

THOVERT,  J.  189,  Relation  entre  la  diffusion  et  la  viscosite.  Compt. 
rend.  138,  482  (1904);  1  p. 

THRELFALL.  Motion  of  Gases  in  Pipes.  Proc.  Inst.  Mech.  Engineers 
(1904). 

THURSTON,  R.  H.  (1)  Friction  and  Lost  Work.  Wiley  &  Sons  (1898);  6 
ed.,  380  pp.;  (2)  Friction  and  Lubrication.  Am.  Assoc.  Adv.  Sci. 
61  (1878);  (3)  Friction  and  Lubrication.  Railroad  Gazette  Pub.  Co. 
(1879);  (4)  Wagn.  Jahrb.  833  (1880). 

TISSOT,  R.     Viscosity  of  Blood.     Folia  haematol.  4,  #4  (1907). 

TODHUNTER  and  PEARSON.  A  History  of  the  Elasticity  and  Strength  of 
Materials.  3  Vols. 

TOKAR,  E.  Versuche  tiber  den  zeitlichen  Verlauf  der  Viscositatsander- 
zungen  bei  Colloidgemischen.  Physik.  Z.  14,  591  (1913);  Diss.  Zurich 
(1909);  37  pp. 

TOMLINSON,  H.  242,  (1)  The  Coefficient  of  Viscosity  of  Air.  Phil.  Trans. 
London  177,  767  (1886);  19  pp.;  Cp.  Proc.  Roy.  Soc.  London  40,  40 
(1886);  (2)  Do.,  Appendix.  Proc.  Roy.  Soc.  London  41,  315  (1886); 
2  pp. ;  (3)  The  Influence  of  Stress  and  Strain  on  the  Physical  Properties 
of  Matter.  Part  I.  Elasticity.  The  Internal  Friction  of  Metals. 
Proc.  Roy.  Soc.  London  40,  240  (1886);  2  pp.;  (4)  Do.,  Part  I.  Elas- 
ticity. The  Effect  of  Magnetization  on  the  Elasticity  and  Internal 
Friction  of  Metals.  Proc.  Roy.  Soc.  London  40,  447  (1886);  2  pp.; 
(5)  Do.,  Part  I.  The  Effect  of  Change  of  Temperature  on  the  Internal 
Friction  and  Torsional  Elasticity  of  Metals.  Proc.  Roy.  Soc.  40, 
343  (1886);  2  pp.;  (6)  On  Certain  Sources  of  Error  in  Connection  with 
Experiments  on  Torsional  Vibrations.  Proc.  Phys.  Soc.  8,  90  (1887);  5 
pp. 

TOWER,  B.  263,  278,  Report  on  Friction  Experiments.  Proc.  Inst.  Mech. 
Eng.,  632  (1883);  27pp.;  Do.,  29  (1884);  7pp.;  Do.,  Ill  (1891);  Tower's 
Verfahren  und  Apparat  zur  Priifung  von  Schmiermitteln.  Dingler's 
Poly  tech.  J.  252,  12;  5  pp.;  Do.,  255,  129. 

TRAUBE,  J.     58,  121,  123,  (1)  Ber.  19,  871  (1886);  (2)  The  Viscostagono- 


422  INDEX 

meter.  Methods  for  Determination  of  Surface  Tension,  Viscosity  and 
Adsorption.  Biochem.  Z.  42,  500  (1912);  3  pp.;  (3)  On  the  Influence 
of  Viscosity  and  Surface  Tension  on  Biological  Phenomena.  Internat. 
Z.  physik.-chem.  Biol.  1,  260  (1914). 

TRAUTZ,  M.  and  HENNING,  H.  Die  Winklersche  Beziehung  zwischen 
innere  Reibung  und  Gasabsorption.  Z.  physik.  Chem.  67,  251  (1907); 
4pp. 

TREITSCHKE,  W.  (1)  Die  innere  Reibung  des  geschmolzeneu  Schwefels. 
Z.  physik.  Chem.  58, 433  (1907); 2  pp.;  Cp.  Beck;  (2)  tTber  die  Charakter- 
isiening  von  Schmelzflussen  mit  Hilfe  der  Konstanten  der  inneren 
Reibung.  Diss.  Leipzig  (1905);  48  pp. 

TRESCA,  H.  235,  (1)  M6moire  sur  1'ecoulement  des  corps  solides.  Me"m. 
Pro's,  a  1'Acad.  de  1'Institut  de  France  18,  773  (1868);  67  pp.;  (2) 
Compt.  rend.  60,  1197  (1869);  4  pp.;  (3)  Memoire  sur  l'6coulement  des 
corps  solides.  M6m.  pr6s.  a  1'Acad.  de  1'Institut  de  France  20,  75 
(1872);  61  pp.;  (4)  Complement  au  Me"moire  sur  l'6coulement  des  corps 
solides.  Me"m.  pro's,  a  1'Acad.  de  1'Institut  de  France  20,  281  (1872); 
8  pp.;  (5)  Me"moire  sur  le  poinconnage  des  me"taux.  M6m.  pr6s.  a 
1'Acad.  de  1'Institut  de  France  20,  617  (1872);  212  pp.;  (6)  Me'moire 
comple'mentaire  sur  le  poingonnage  des  m6taux.  Me"m.  pr<5s.  a  1'Acad. 
de  1'Institut  de  France  20,  829  (1872);  (7)  On  Further  Applications  of 
the  Flow  of  Solids.  Proc.  Inst.  Mech.  Eng.  301  (1878);  45  pp. 

TROMMSDORF,  F.  Untersuchungen  iiber  die  innere  Reibung  des  Blutes  und 
ihre  Beziehung  zur  Albanese'sche  Gummilosung.  Arch.  f.  exp.  Path, 
u.  Pharm.  45,  66  (1901). 

TROUTON,  F.  7,  226,  227,  228,  237,  On  the  Coefficient  of  Viscous  Traction 
and  its  Relation  to  that  of  Viscosity.  Proc.  Roy.  Soc.  London  (A) 
77,426  (1906);  14pp. 

TROUTON,  F.  and  ANDREWS,  E.  7,  218,  On  the  Viscosity  of  Pitch-like 
Substances.  Phil.  Mag.  (6)  7,  347  (1904);  9  pp. 

TRUMPP.  Viscositat,  Hamoglobin  und  Eiweissgehalt  des  kindlichen 
Blutes.  Munch.  Med.  Wochschr.  #42  (1909). 

TASAKALOTOS,  D.  97,  (1)  Die  innere  Reibung  in  kritischen  Zone. 
Z.  physik.  Chem.  68,  32  (1909);  6  pp.;  (2)  Sur  la  viscosite"  des  melanges 
binaires  des  composes  organiques.  Formation  de  combinaisons 
mole-culaires  a  I'Stat  liquide.  Bull.  soc.  chim.  (4)  3,  234  (1908);  (3) 
tJber  die  Viscositat  binarer  Gemische  organischen  Verbindungen. 
Bull.  soc.  chim.  (4)  3,  242  (1908);  (4)  Bull.  soc.  chim.  (4)  6,  234,  397 
(1903);  7  pp. 

TSCHWEVSKY.  (1)  Geschwindigkeit  und  Widerstand  in  der  Strombahn  der 
Arteria  carotis  und  cruralis  ecc.  Arch.  f.  d.  Ges.  Physiol.  97,  210 
(1903);  (2)  Contribution  a  1'^tude  de  la  distribution  sanguine  dans  les 
vaisseaux  nourriciers  du  coeur.  Rousski  Vratch  441  (1904). 

TURPIN,  S.  S.  and  WARRINGTON,  A.  W.  Apparent  Viscosity  of  Ice.  Phil. 
Mag.  18,  120  (1884). 

UBBELOHDE,  L.     324,  (1)  Ehrenrettung  des  Kranzbrenners  des  Englerschen 


INDEX  423 

Schmierolviskosimeters.  Chem.  Ztg.  31,  28  (1907);  1  p.;  (2)  Tabellen 
zum  Englerschen  Viskosimeter.  Verlag.  S.  Hirzel,  Leipzig  (1907); 
28  pp.;  (3)  Petroleum  4,  Heft  15,  841;  (4)  Einige  Neuerungen  am  Engler- 
schen Schmierolviskosimeter  und  Tabellen  ftir  das  Viskosimeter. 
Chem.  Zth.  31,  38  (1907);  2  pp.;  (5)  Theorie  der  Reibung.  S.  Hirzel, 
Leipzig;  (6)  Theory  of  Friction  of  Lubricated  Machinery.  Petroleum 
7,  938  (1912);  Seifensieder  Ztg.  39,  1009-10,  1045^6;  Stahl  und  Eisen 
32,  1685  (1912);  5  pp.;  (7)  The  Theory  of  Lubrication.  Petroleum 
Rev.  27,  293,  325  (1913);  3  pp.;  Cp.  General  Electric  Rev.  18,  968 
(1915). 

UBBELOHDE  und  AGTHE.  Diplomarbeit,  Karlsruhe  (1912);  Cp.  Engler 
Erdol,  p.  53,  54,  55,  56. 

UBBELOHDE,  L.  &  HOFSASS,  M.  A  New  Gas  Meter,  the  "Capomesser;" 
and  a  Viscometer  for  Gases.  J.  Gasb.  55,  557  (1912);  4  pp.  Z. 
Elektrochemie  19,  32  (1912);  3  pp. 

UCHIYAURA,  K.  Uber  Viskositatsbestimmungen  der  Milch  und  der  geb- 
rauchlichsten  Sauglingsnahrungen.  Diss.  Miinchen  (1909);  38  pp. 

ULMER,  A.  Die  Bestimmung  des  Volumes  der  Blutkorperchen  auf  viscosi- 
metrischem  Wege.  Diss.  Zurich  (1909);  29  pp. 

ULTEE,  A.  J.  (1)  Results  of  Viscosity  Determinations  of  Rubber  Solutions. 
Arch.  Rubbercult.  2,  331  (1918);  16  pp.;  (2)  Viscosity  Determinations 
and  Uniformity.  Do.  3,  24  (1919);  12  pp. 

UMANI,  A.  SuU'attrito  interno  del  mercuric.  Cim.  (4)  3,  151  (1896); 
16pp. 

UPTON,  G.  B.  The  Properties  of  Oils  and  their  Relation  to  Lubrication. 
Sibley  J.  of  Engineering  30,  277  (1915);  7  pp. 

VALENTA,  E.     6,  Ein  einfacher  Apparat  zur  Bestimmung  der  Zahfliissigkeit 

von  Firnissen.     Chem.  Ztg.  30,  583  (1906);  1  p. 
VALENTI,  A.     L'influenza  della  viscositS,  sul  comportamento  delle  soluzioni 

saline  verso  il  protoplasma  vegetale  e  animale.     Arch.  di.  farmacol. 

sperim.  e.  scienze  affini  3,  492  (1904). 
VAN  DER  BELLEN,  E.     Uber  eine  neue  Methode  der  Bestimmung  der  Plasti- 

zitat  der  Tone.     Chem.  Zeit.  27,  433  (1903);  1  p. 
VATJTIER,  T.     (1)  Sur  la  vitesse  d'ecoulement  des  liquides.     Compt.  rend. 

102,  165  (1886);  2  pp.;  Cp.  Phil.  Mag.  (5)  21,  285  (1886);  2  pp.;  (2) 

Do.     Compt.  rend.  103,  372  (1886);  4  pp. 
VEAZEY,  W.     183,  The  Conductivity  and  Viscosity  of  Solutions  of  Certain 

Salts  in  Water,  Methyl  Alcohol,  Ethyl  Alcohol,  Acetone,  and  Binary 

Mixtures  of  these  Solvents.     Diss.  Johns  Hopkins  (1907);  49  pp.     Cp. 

Jones. 
VEINBERG,  B.  P.     (1)  Study  of  Substances  Having  High  Viscosity.     J.  Russ. 

Phys.  Chem.  Soc.  Phys.  Pt.  44,  1  (1912);  1  pp.;  (2)  Contributions  to  the 

Study   of   Substances   Having  Large   Coefficients   of   Viscosity.     III. 

Influence  of  Temperature   on   the   Viscosity   of  Pitch   and   Asphalt. 

Do.,  44,  201  (1913);  29  pp.;  IV.  Resistance  of  a  Viscous  Medium  to  the 

Motion  of  a  Solid  Body.     Do.  44,  241    (1913);  11  pp.;  V.  Further 


424  INDEX 

Experiments  on  the  Flow  of  a  Viscous  Liquid  in  a  Canal.  Do.,  (3) 
44,  252  (1913);  5  pp.;  (3)  Phenomena  in  Liquids  under  Homogeneous 
Friction.  Do.,  44,  514  (1913);  2  pp.;  (4)  Supplement  to  the  Article 
of  S.  I.  Monstrov.  Do.,  44,  503  (1913);  1  p.;  (5)  Inner  Friction  of 
Binary  Systems.  Do.  46,  701  (1913);  5  pp.;  (6)  Some  Methods  for 
Studying  the  Viscosity  of  Solids.  Proc.  Physic.  Soc.  London  19, 
472  (1904);  3  pp.;  (7)  tTber  die  innere  Reibung  des  Eises.  Ann. 
Physik.  (4)  18,  81  (1905). 

VEINBERG,  B.  and  SMIRNOV,  I.  Comparison  of  some  Methods  of  Deter- 
mining the  Viscosity  of  Pitch.  J.  Russ.  Phys.  Chem.  Soc.  (Phys.  Pt.) 
44,  3  (1912);  32  pp. 

DE  ST.  VENANT,  BARRE.  6,  (1)  Rapport  sur  un  M6moire  de  M.  Kleintz 
intitul6  "Etudes  sur  les  forces  moleculaires  dans  les  liquides  en  mouve- 
ment  et  application  a  1'Hydrodynamique."  Compt.  rend.  74,  1847; 
25  pp.;  (2)  Sur  Fhydrodynamique  des  cours  d'eau.  Do.,  74,  570,  649, 
693,  770;  (3)  Sur  1'intensite1  des  forces  capables  de  dcformer,  avec  con- 
tinuites  des  blocs  ductile  etc.  Do.,  74,  1009. 

VERNON,  H.  M.  The  conditions  of  action  of  "trypsin"  on  fibrin.  J. 
Physiol.  26,  405  (1900);  21  pp. 

VERSCHAFPELT,  J.  E.  The  Viscosity  of  Liquefied  Gases.  VI.  Observations 
on  the  Tortional  Oscillatory  Movement  of  a  Sphere  in  a  Viscous 
Liquid  with  Finite  Angles  of  Deviation,  etc.  Proc.  Acad.  Sci.  Amster- 
dam 19,  1062  (1917);  12  pp.;  VII.  Torsional  Oscillatory  Motion  of  a 
Body  of  Revolution  in  a  Viscous  Liquid.  Do.,  6  pp.;  VIII.  The  Simi- 
larity in  the  Oscillatory  Rotation  of  a  Body  of  Revolution  in  a  Viscous 
Liquid.  Do.,  5  pp.;  IX.  Preliminary  Determination  of  the  Viscosity 
of  Liquid  Hydrogen.  Do.,  4  pp. ;  X.  The  Viscosity  of  Liquid  Hydrogen. 
Do.,  20,986  (1918);  4  pp. 

VESELY,  V.     The  Viscosity  of  Glass.     Sprechsaal  44,  441  (1911);  7  pp. 

VICAT,  L.  G.  A  Practical  and  Scientific  Treatise  on  Calcareous  Mortars  and 
Cements.  J.  F.  Smith's  Trans.  London  (1837). 

VILLARI,  VON.  Sull'efflusso  del  mercuric  per  tubi  di  vetro  di  piccolo  diametro. 
Mem  dell' Ace.  dello  Institute  di  Bologna  (3)  6,  (1876). 

DE  VILLEMONTIE,  G.  192,  Encyclopedic  des  Aide-M6moire,  Section  de 
I'lng&ueur.  Resistance  electrique  et  Fluidity.  Gauthier-Villars;  188 
pp. 

VOGEL,  H.  242,  247,  The  Viscosity  of  certain  Gases  and  the  Variation  with 
Temperature  at  Low  temperatures.  Ann.  Physik.  (4)  43,  1235  (1914); 
37pp. 

VOGEL,  COLEMAN,  and  FISHER.  Untersuchungen  der  Mineralole  und 
Fette.  Holde  2d  Ed.  (1905);  Springer,  p.  100. 

VOIGT,  W.  (1)  Tiber  innere  Reibung  fester  Korper  insbesondere  der 
Metalle.  Wied.  Ann.  47,  671  (1897);  23  pp.;  Cp.  Abhandl  der  konigl. 
Gesell.  der  Wissens.  zu  Gottingen  36  (1890);  and  38  (1892);  (2)  Einige 
Beobactungen  iiber  Elastizitat  und  innere  Reibung  von  Legierungen 
aus  Kadmium  und  Zink,  angestellt  von  J.  Miller.  Physik.  Z.  9,  256 
(1906);  2pp. 


INDEX  425 

VOLLMER,  B.  194,  Die  elektrische  Leitfahigkeit  von  einiger  Salzen  in 
Xthyl  und  Methyl  Alkohol.  Wied.  Ann.  52,  347  (1894);  29  pp.; 
Cp.  Festschrift  des  Realgymnasiums  der  Frankeschen  Stiftung  zu 
Halle  (1894). 

VORIS,  O.  E.     (Sweet  milk  as  a  cutting  fluid).     Machinery  22,  64  (1916). 

VRIENSNIEWSKI.     J.  Soc.  Phys.  Chem.  St.  Pet.  43,  1383  (191-). 

WAGNER,  J.  3, 128,  179,  184,  186,  (1)  t)ber  die  Zahigkeit  von  Salzlosungen. 
Wied.  Ann.  18,  259  (1883);  33  pp.;  (2)  tTber  die  innere  Reibung  ver- 
diinnter  Salzlosungen.  Z.  physik.  Chem.  5,  31  (1890);  22  pp. 

WAGNER,  J.  and  MUHLENBEIN,  J.  tJber  die  innere  Reibung  von  Losungen. 
Z.  physik.  Chem.  46,  867  (1903);  11  pp. 

WALDEN,  P.  179,  194,  (1)  Relation  between  Molecular  Conductivity  and 
Viscosity  in  Non-Aqueous  Solutions.  Bull.  Acad.  Imp.  Sci.  St.  Peters- 
burg, 559  (1913);  23  pp.;  (2)  The  Relation  between  the  Limit  of  Molecu- 
lar Conductivity  and  Viscosity.  Z.  physik.  Chem.  78,  257  (1912); 
26  pp.;  (3)  Trans.  Faraday  Soc.  6,  75  (19—);  (4)  Organische  Losungs- 
und  lonizierungsmittel.  III.  Innere  Reibung  und  der  Zusammenhang 
mit  der  Leitvermogen.  Z.  physik.  Chem.  66,  207  (1906);  Cp.  Z. 
Elektrochem.  12,  77  (1906);  1  p. 

WALKER,  W.  J.  The  Relationship  between  the  Viscosity,  Density  and 
Temperature  of  Salt  Solutions.  Phil.  Mag.  (6)  27,  288  (1914);  9  pp. 

WARBURG,  E.  32,  237,  (1)  Uber  die  Dampfung  der  Tone  fester  Korper 
durch  innere  Widerstande.  Berl.  Monatsber.  538  (1869);  12  pp.; 
Pogg.  Ann.  139,  89  (1870);  15  pp.;  (2)  tlber  den  Ausfluss  des  Queck- 
silbers  aus  glasernen  Capillarrohren.  Pogg.  Ann.  140,  367  (1870); 
(3)  tJber  die  Gleitung  der  Case  an  Glaswanden.  Pogg.  Ann.  159, 
399  (1876);  17  pp.;  (4)  Uber  das  Gleichgewicht  eines  Systems  ausge- 
dehnter  Molecule  und  die  Theorie  der  elastischen  Nachwirkung.  Wied. 
Ann.  4,  232  (1878);  17  pp.;  (5)  Uber  die  Torsion.  Wied.  Ann.  10, 
13  (1880);  22  pp.;  (6)  Magnetische  Untersuchungen.  I.  tlber  einige 
Wirkungen  der  Coercitivkraft.  Wied.  Ann.  13,  141  (1881);  24  pp. 

WARBURG,  E.  and  BABO,  L.  VON.  138,  244,  (1)  tJber  eine  Methode  zur 
Untersuchung  der  gleitenden  Reibung  fester  Korper.  Wied.  Ann. 
2,  406  (1877);  12  pp.;  (2)  tlber  den  Zusammenhang  zwischen  Viscositat 
und  Dichtigkeit  bei  fliissigen  inbesonders  bei  gasformig  fltissigen  Kor- 
pern.  Wied.  Ann.  17,  390  (1882);  37  pp.;  Cp.  Ber.  liber  Verhand- 
lungen  der  naturforschenden  Gesellschaft  zu  Freiburg  8,  1  (1862); 
44pp. 

WARBURG,  E.  and  SACHS,  J.  138,  140,  t)ber  den  Einfluss  der  Dichtigkeit 
auf  die  Viscositat  tropf barer  Fliissigkeiten.  Diss.  Freiburg;  Wied. 
Ann.  22,  518  (1884);  5  pp. 

WASHBURN,  E.  W.  196,  197,  (1)  The  Laws  of  "Concentrated"  Solutions: 
II.  The  Estimation  of  the  Degree  of  lonization  of  Electrolytes  in 
Moderately  Concentrated  Solutions.  J.  Am.  Chem.  Soc.  33,  1461 
(1911);  18  pp.;  Cp.  Tech.  Quarterly  21,  2023  (1908);  (2)  A  Factory 
Method  for  Measuring  the  Viscosity  of  Pot-Made  Glass  during  the 


426  INDEX 

Process  of  Manufacture,  together  with  Some  Discuss  on  of  the  Value 
of  Viscosity  Data  to  the  Manufacturer.  J.  Am.  Ceram.  Soc.  3,  735 
(1920);  15  pp.;  (3)  Physical  Chemistry,  2d.  ed.,  McGraw-Hill 
Book  Co. 

WASHBURN,  E.  W.  &  MAC!MES,  D.  A.  The  Laws  of  "Concentrated" 
Solutions.  III.  The  lonization  and  Hydration  Relations  of  Elec- 
trolytes in  Aqueous  Solution  at  0°C.  J.  Am.  Chem.  Soc.  33,  1687 
(1911);  28  pp. 

WASHBURN,  E.  W.  &  WILLIAMS,  G.  Y.  (1)  Precision  Viscometer  for 
Measurement  of  Relative  Viscosity  and  the  Relative  Viscosities  of 
Water  at  0°,  18°,  25°,  and  50°.  J.  Am.  Chem.  Soc.  35,  737  (1913); 
33  pp.;  (2)  The  Viscosities  and  Conductivities  of  Aqueous  Solutions 
of  Raffinose.  J.  Am.  Chem.  Soc.  36,  750  (1913);  4  pp. 

WATSON,  F.  7,  256,  Viscosity  of  Liquids  as  determined  by  measurement 
of  Capillary  Waves.  Physic.  Rev.  16,  20  (1902);  19  pp. 

WAY,  J.  T.  On  the  Power  of  Soils  to  absorb  Manure.  Roy.  Agric.  Soc. 
J.  11  (1850);  66  pp. 

WEBB,  J.  The  Viscous  Dynamometer.  Science  (N.  S.)  16,  338  (1902);  2 
pp. 

WEBER,  F.  (1)  Plasma  Viscosity  of  Plant  Cells.  Z.  allegemein  Physiol. 
97,  1  (1918);  20  pp.;  (2)  Viscometry  of  Living  Protoplasm.  Kolloid- 
Z.  20,  169  (1917);  4pp. 

WEBER,  W.  237,  (1)  Vorlesung  de  fili  bombycini  vi  elastica.  Gotting. 
Gelehrt.  Anz.,  St.  8,  65  (1835);  12  pp.;  (2)  Uber  die  Elasticity  der 
Seidenfaden.  Pogg.  Ann.  34,  247  (1835);  11  pp.;  (3)  Pogg.  Ann.  34, 
1  (1841);  Cp.  Comm.  Soc.  Gottingen  3,  45  (1841). 

WEINBERG.     Cp.  Veinberg. 

WEINSTEIN,  M.  B.  The  Internal  Friction  of  Gases.  I.  The  First  Coeffi- 
cient of  Friction.  Ann.  Physik.  50,  601  (1916);  53  pp.;  Do.,  II.  The 
Second  Coefficient  of  Friction,  the  Thermodynamic-Hydrodynamic 
Equations  of  G.  Kirchhoff,  and  Maxwell's  Gas  Theory.  Do.,  60.  796 
(1916);  18pp. 

WEISBACH.  18,  Lehrbuch  der  Ingenieur  und  Machinenmechanik.  Experi- 
mental hydraulik,  3d.  Ed.  1,  736. 

WELLS,  H.  M.  and  SOUTH  COMBE,  J.  E.  Theory  and  Practice  of  Lubrication: 
"Germ"  Process.  J.  Soc.  Chem.  Ind.  39,  51  (1920);  9  pp.;  Cp.  South- 
combe  Petroleum  Times  3,  173,  201  (1920);  4  pp. 

WELSH,  W.  N.     Viscosity  of  the  Blood.     Heart  3,  112  (1912);  19  pp. 

WENDRINER,  M.  Ein  einf aches  Viscosimeter.  Z.  f.  angew  Ch.  545  (1894); 
2pp. 

WENDT,  P.  Reply  to  Ubbelohdes's  Article  "The  Theory  of  the  Friction 
of  Lubricated  Machine  Parts."  Petroleum  8,  678  (1913);  8  pp. 

WERIGIN,  N.,  Lewkojeff,  J.,  and  TAMMANN,  G.  236,  tJber  die  Ausfluss- 
geschwindigkeit  einiger  Metalle.  Ann.  Physik.  (4)  10,  647  (1903); 
8  pp.  Cp.  Tammann. 

WEST,  G.  D.  The  Resistance  to  the  Motion  of  a  Thread  of  Mercury  in  a 
Glass  Tube.  Proc.  Roy.  Soc.  London  (A)  86,  25  (1910);  11  pp. 


INDEX  427 

WETZSTEIN,  G.  68,  tTber  Abweichungen  von  Poiseuilleschen  Gesetz. 
Diss.  Munchen  (1899);  Wied.  Ann.  68,  441  (1899);  30  pp. 

DE  WHALLET,  H.  C.  and  SIEGFRIED.  A  Gravimetric  Method  of  Comparing 
Viscosities  of  Varnish,  etc.  Analyst  44,  288  (1919) ;  1  p. 

WHEELER.  Clay  Deposits.  Chap.  V.  Plasticity  of  Clay.  Mo.  Geol. 
Survey  11,  97  (1896);  17  pp. 

WHETHAM,  W.  31,  32,  213,  (1)  On  the  Alleged  Slipping  at  the  Boundary 
of  a  Liquid  in  Motion.  Proc.  Roy.  Soc.  London  48,  225  (1890);  (2) 
On  the  Velocity  of  Ions.  Phil.  Trans.  (A)  186,  507  (1896);  16  pp. 

WHITE,  G.  F.  97,  104,  (1)  Study  of  the  Viscosity  of  Fish  Oils.  J.  Ind. 
Eng.  Chem.  4,  106  (1912);  4  pp.;  (2)  Fluidity  of  Fish  Oils  as  an  Additive 
Property.  Do.  4,  267  (1912);  3  pp.;  (3)  Ein  Neues  Viscosimeter  und 
seine  Anwendung  auf  Dlut  und  Blutserum.  Biochem.  Z.  37,  482 
(1911);  7  pp.;  Cp.  Bingham. 

WHITE,  G.  F.  &  THOMAS,  A.  Studies  on  Fish  Oils.  III.  Properties  of 
Fish  and  Vegetable  Oil  Mixtures.  J.  Ind.  and  Eng.  Chem.  4,  878 
(1912);  5  pp. 

WHITE,  G.  F.  &  Twixnra,  R.  H.  (1)  The  Fluidity  of  Butter  Fat  and  its 
Substitutes.  J.  Ind.  Eng.  Chem.  6,  .568  (1913) ;  5  pp. ;  (2)  The  Viscosity 
of  Undercooled  Water  as  Measured  in  a  New  Viscosimeter.  Am.  Chem. 
J.  60,  380  (1913);  9  pp. 

WEECHERT,  E.  237,  tTber  elastische  Nachwirkung.  Diss.  Konigsberg 
(1889);  64  pp. 

WIEDEMANX,  E.  79, 246,  (1)  Arch.  sci.  phys.  nat.  66, 273  (1876) ;  (2)  tlber  die 
Beziehung  zwischen  dem  Reibungs  und  Leitungs-widerstand  der 
Losungen  von  Salzen  in  verschiedenen  Losungsmitteln.  Wied.  Ann. 
20,537  (1883);  2  pp. 

WIEDEMAXN,  G.  &  VERDET.  2,  6,  192,  M£moire  sur  le  mouvement  des 
liquides  qui  s'observe  dans  le  circuit  de  la  pile  voltaique  et  au  les  rela- 
tions de  ce  mouvement  avec  1'electrolyse.  Ann.  de  chim.  et.  de  phys. 
(3)  62,  224  (1858);  30  pp.;  Extraits  par  Verdet  Pogg.  Ann.  99, 
77  (1856). 

WLTKANDEB.  81,  92,  Lunds  Physiogr.  Sallsk.  Jubelskr.  Lund  (1878); 
Wied.  Beibl.  3,  8  (1879). 

WILBERFORCE,  L.  17,  On  the  Calculation  of  the  Coefficient  of  Viscosity 
of  a  Liquid  from  its  Rate  of  Flow  through  a  Capillary  Tube.  PhiL 
Mag.  (5,  31,  407  (1891);  8  pp. 

WILKIXS.     Elektrotechn.  Zeitschr.  26,  135  (1904). 

WILLERS,  F.  Viscosity  Anomalies  of  Emulsions  in  the  Conditions  of 
Turbulent  Flow.  Physik.  Z.  10,  244  (1908);  4  pp. 

WILSOX,  H.  A.  7,  190,  191,  On  the  Velocity  of  Solidification  of  Super- 
cooled Liquids.  Proc.  Camb.  PhiL  Soc.  10,  I,  25;  Phil.  Mag.  60,  238 
(1900);  13pp. 

WI.NKELMA.VN.    Handbuch  der  Physik.  578-582  (1891).    Cp.  Graetz  &  Jager. 

WINKLER,  L.  Gesetzmassigkeit  bei  der  Absorption  der  Gase  in  Flussig- 
keiten.  Z.  physik.  Chem.  66,  171  (1992);  13  pp.;  Cp.  2.  physik.  Chem. 
10,  144  (1892);  (2)  Do.,  Z.  physik.  Chem.  66,  344  (1906);  11  pp. 


428  INDEX 

WOLFF,  H.  (1)  The  Determination  of  the  Viscosity  of  Varnishes.  Farben. 
Ztg.  17,2108;  (2)  Beitrag  zurKenntniss  der  Leitf ahigkeiten  gemischter 
Losungen  von  Elektrolyten.  Z.  physik.  Chem.  40,  222  (1902); 
34pp. 

WOUDSTRA,  H.  205,  Uber  die  innere  Reibung  kolloidaler  Silberlosungen 
Z.  physik.  Chem.  63,  619  (1908);  4  pp.;  Cp.  Chem.  Weekblad  6.  303, 
602;  (2)  The  Degree  of  Dispersion  and  Viscosity.  Z.  Chem.  Ind. 
Kolloide  8,  73  (1911);  8  pp.;  (3)  The  Viscosity  and  Coagulation  of 
Caoutchouc  Solutions.  Z.  Chem.  Ind.  Kolloide  6,  31  (1909);  2  pp.; 
(4)  Kolloid.-Z.  8,  73  (1911). 

WRIGHT.     105.     C.  Kendall  and  Wright. 

WROBLEWSKI,  S.  VON.  (1)  tlber  die  Abhangigkeit  der  Constante  der 
Verbreitung  der  Gase  in  einer  Fltissigkeit  von  der  Zahigkeit  der 
letztern.  Wied.  Ann.  7,  11  (1879);  13  pp.;  (2)  tJber  die  Natur 
der  Absorption  der  Gase.  Wied.  Ann.  8,  29  (1879);  24  pp. 

WULLNER.     Lehrbuch  der  Experimental  physik.  4th  Ed.,  259  (1882). 

YEN,  KIA-LOK.  An  Absolute  Determination  of  the  Coefficients  of  Viscosity 
of  Hydrogen,  Nitrogen,  and  Oxygen.  Phil.  Mag.  38,  582  (1919);  16 
pp. 

ZAHM,  A.  Atmospheric  Friction  on  Even  Surfaces.  Phil.  Mag.  (6)  8, 
58  (1904);  9  pp. 

ZAKRZEWSKTEGO,  K.  O  oscylacyi  krazka  w  plynie  lepkin.  Rozprawy 
Akademii  (A)  42,  392  (1902);  7  pp. 

ZANDA,  G.  B.  286,  (1)  Viscosity  of  the  Blood  During  the  Absorption  of 
Glucose.  Arch.  Ital.  Biol.  62,  79  (1910);  4  pp.;  Zentr.  Biochem.  Bio- 
phys.  10,  1006;  (2)  Azione  dei  farmaci  sulla  digestione  pepsinica  dal 
punto  di  vista  fisico-chimico.  Giornale  della  R.  Ace.  di.  Torino  10, 
#7  and  8  (68). 

ZAREMBA,  S.  Krakauer  Anz.  380,  403  (1903);  Rozpr.  Akad.  (A)  43,  14 
(1904);  7  pp.;  Krakauer  Anz.  86  (1903);  8  pp. 

ZEMPLEN,  G.  (1)  Bestimmung  des  Koefficienten  der  inneren  Reibung  der 
Gase  nach  einer  neuen  experimentalen  Methode.  Ann.  Physik.  (4) 
19,  783  (1906);  23  pp.;  (2)  Do.,  Ann.  Physik.  (4)  29,  869  (1909); 
39  pp.;  Cp.  Math,  natro.  Ber.  Ungarn  19,  74  (1904;;  7  pp.  and  Math. 
Termt.  Ert.  Budapest  23,  561  (1905);  (3)  Investigations  on  the  Vis- 
cosity of  Gases.  Ann.  physik.  38,  71  (1912;;  54  pp. 

ZERI,  A.  (1)  La  viscosita  della  bile  umana.  Arch,  di  farmac.  speriment.  e 
scienze  affini  4,  279  (1905);  (2)  Un  Nuovo  carattere  differenziale  tra 
essudati  e  trasudati.  II  Politecnico.  Sezione  pratica  12,  1373  (1905). 

ZEUNER.     18,  Civilingenieur  1,  84. 

ZIMMER,  O.  (1)  The  Viscosity  of  Ethylene  and  Carbon  Monoxide  and 
its  Changes  at  Low  Temperatures.  Ber.  deut.  physik.  Ges.  471 
(1912). 

ZOJA,  L.  Physikalisch-chemische  Untersuchung  der  Reaktionen  zwischen 
Eiereiweiss  und  Essigsaure.  Roll.  Z.  3,  249,  269  (1908) ;  20  pp. 


INDEX  429 

ZOLLEK,  H.  F.  (1)  The  Viscosity  of  Casein  Solution.  I.  The  Effect  of  P//. 
Science  60,  49  (1919);  (2)  Casein  Viscosity  studies.  J.  Gen.  Physiol. 
3,635  (1921);  16pp. 

ZSCHOKKE,  B.  (1)  Untersuchungen  tiber  die  Plastizitat  der  Thone.  Bull. 
Soc.  d'Encouragement  d'Industrie  Nationale  103,  619  (1902);  40  pp.; 
(2)  Untersuchungen  tiber  die  Bildsamkeit  der  Thone.  Baummaterial- 
ienkunde  7,  377,  393  (1902);  7  pp.;  (3)  Untersuchungen  iiber  die  Plas- 
ticitat  der  Thone.  Do.,  #1,  2,  3,  4,  5,  6  (1903);  18  pp. 

ZUR  NEDDEN,  F.  Induced  Currents  of  Fluids.  Proc.  Am.  Soc.  Civ.  Engi- 
neers 41,  1351  (1915);  54  pp. 


SUBJECT  INDEX 


Decimals  indicate  the  location  of  reference  on  the  page.     Two  or  more 
references  on  the  same  page  are  indicated  by  a  +  sign. 


Absorption  235,  259,  427.9,    428.3 
Acetic  anhydride,  410.7 

acid,  402.2 
Acetylene,  403.6 
Acids,  aliphatic,  115 
Additivity  of  fluidities,  82,  83,  104, 

412.8 
Adhesion,   31,   221,   230,   257,   261, 

268,  274 

Adsorption,  378.9,  394.2 
Adulteration,  370.4 
Air,    361.5,    369.2,    373.5+,    374.2, 

376.1,  382.5+,       383.1, 
386.4,  396.1,  399.4,  402.6, 
407.9,  409.1,  410.1,  414.3, 

415.2,  421.6 

Albumen,  404.6,  416.4,  419.1,  428.9 
Alcohols,  116,  177,  349.7 
Alloys,  349.4,  361.6,  424.9 
Aluminium  hydroxide,  372.1,  387.2 

oleate  in  oil,  406.8 
Amalgams,  415.4      • 
Amides,  370.1 

Amines,  366.5,  368.1,  371.3,  420.7 
Ammonia,  371.3 
Ammonium  iodide,  186 

nitrate,  180,  356.2,  374.6 

thiocyanate,  374.6 
Aniline,  416.2  + 

Anisotropic  liquids,  96,  209,  356.4, 
359.6,  367.7,  390.4,  392.7, 

401.6,  405.8,  407.7,  413.5 
Annealing,  212 

Antifriction  metals,  278 
Antimony  chloride,  391.2 
Antipyrine  solutions,  414.9 
Argon,  364.5,  389.8,  408.7,  409.3 +, 

413.7,  414.8,  420.2 


Asphalt,  365.3,  399.5+,  410.5,  423.9 
-base     oils     vs.      paraffin-base 

oils,  274 
Association,  92,  112,  119,  161,  184, 

378.1,  415.7,  420.6 
Atomic    constants,    108,    111,    126, 

144,  186,  196 
diameters,  253 
weights  and  v.  of  gases,  250 
Avogadro's  constant,  253 


B 


Barium  sulphate,  349.8 

Bath,  307 

Beater-stock,  360.1 

Belting,  283 

Bent  capillaries,  376.6 

Benzene,  354.6,  366.3,  381.8 

Benzyl  benzoate,  354.6 

Bile,  359.8,  428.8 

Binders,  sticking  strength  of,  347.2 

Biology,  284 

Blood,  284,  348.1,  349.3,  350.5, 
352.6+,  353.1,  355.6+, 
356.9,  359.3+,  360.2  +  , 

361.8,  362.3+,      363.9, 
364.1,       365. 1+,       367.8, 
368.4+,  369.4+,  370.6 +, 
372. 1+,         374.3,      375.2, 

376.9,  379.1,         380.8, 
381.6,       382.4,        383.2+, 
385.1,       386.2+,       387.1, 
389.9.         390.5,         392.2, 
393.1 +.     394.6+,     397.1, 

400.5,  403.4,  404.7,  405.7, 
406.3+,       409.9,       411.6, 

412.6,  413.4,  414.8,  421.5, 
422.6  +  ,     423.4,     426.3+, 
427.3,  428.6 


431 


432 


INDEX 


"Body,"  269 

Body  fluids,  417.3 

Boiling-point,  155 

Brittleness,  403.7 

Bromides,  114 

Bromine,  386.8,  408.8 

Brownian     movement,     188,     190, 

358.7,  415.5 
Bunsen  flame,  352.8 
Butter,  281,  388.1,  427.4 


Cadmium  and  zinc  alloys,  424.9 
iodide,  386.1 

Caesium  nitrate,  398.1 

Calcium  chloride,  416.4 

Calculation  of  fluidity,  314 
of  plasticity,  323 

Caoutchouc,  cp.  rubber. 

Capillarity,  56,  70,  361.5,  370.8, 
406.1,  414.1,  cp.  surface 
tension. 

Capillary  tube  method  for  plas- 
ticity, 222 

Carbon  black,  405.2 

dioxide,  146,  383.2,  405.6,  408.1 
monoxide,  428.9 
tetrachloride  and  benzene,  167 

Carbonyl  sulphide,  359.4 

Casein,  361.4,  419.1,  429.1 

Castor  oil,  363.7,  386.5,  408.4 

Celluloid,  348.3,  415.2 

Cellulose  acetate,  351.1,  402.3 

esters,  351.2,  360.3,  365.2,  373.7, 
403.7,  417.9,  cp.  nitro- 
cellulose. 

Cements,  424.6 

Centipoise,  61 

Ceramics,  286 

Chart  for  conversion,  401.7 

Chemical  composition,  106,  112,  172, 
249,  375.1,  387.6,  390.3, 
407.4,  409.9,  422.7 

Chloral  solutions,  391.1 

Chlorides,  381.3 

Chlorine,  122,  144,  408.8 


Chloroform  and  ether,  175,  364.4 

Chromium  salt  solutions,  370.7, 
374.9,  416.8 

Clay,  221,  229,  281,  349.9,  355.3, 
376.5,  388.4,  391.7,  397.7, 
399.5,  403.6,  408.1,  410.6, 
411.9,  415.4+,  416.5, 
418.9,  429.1 

Close-packing,  228,  229 

Cloud  method,  399.4 

Coagulation,  284,  371.9,  387.2, 
396.8,  411.6,  428.2 

Cohesion,  148,  212,  386.3,  400.1, 
415.3 

Cold  working,  211 

Collisions,  149,  200 

Collisional  viscosity,  147,  151 

Colloidal  solutions,  3,  198,  348.5, 
351.8,  353.7,  360.6,  364.2, 
369.1,  371.8,  372.7, 
374. 1+,  377.2,  378.8, 

379.5,  380.7,         381.4, 
392.9+,     393.2+,     403.9, 
411.6+,       412.8,       413.8, 

414.6,  417.2,   421.6,   428.1 
Colloidoscope,  198 
Colophonium,  52 

Color     of     solutions,      416.3,      cp, 

chromium. 
Comparable      ^temperatures,       115, 

410.4  * 

Compressible  fluids,  49 
Conductivity,    electrical,    191,    192, 

193,    194,    349.1,    349.4  +  * 

353.7,  357.2,  360.6,  364.7, 
367.2+,     368.1,     371.2 +  , 
374.6,        375.6,      376.3+, 
377.4,  379.7,  382.9,  383.4, 
386.6,         389.7,         390.1, 
392.7+,  394.3+,  396.3+, 

399.6,  402.3,  403.2,  405.9, 
406.1,  412.1,  414.1,  415.4, 
416.3,  417.1,  418.5,  423.8, 

424.7,  425.1 +,427.5 
thermal,     252,     358.7,     360.8, 

368.7,  380.3,   390.8,   406.7 
Conjugate  double  bonds,  111 


INDEX 


433 


Consistency,  235,  361.8 
Constants  of  viscometer,  296,  313 
Constitution,  chemical,   121,  352.1, 
354.9,  358.8,  359.1,  367.1, 
366.4+,       372.8,       382.1, 

388.8,  390.7,  405.1,  407.5, 
413.4 

Construction  of  viscometer,  315 
Corresponding  states,  403.4 
Cream,  365.9 
Criterion  of  Reynolds,  40 
Critical-solution    temperature,     94, 

102,  364.8,  372.1 
Critical  state,  365.4,  419.1,  422.7 

velocity  of  flow,  361.9 
Crystalline  liquids,  96;  208,  cp.  anis- 

otropic  liquids. 
Crystallization,    190,    371.8,    372.2, 

375.9,  379.3,  420.2 
Cutting    fluids,     269,    272,     348.5, 

354.3,  420.3,  425.1 
Curcas  oil,  348.9 
Curved  pipes,  368.9 


Deflocculation,  208,  229,  231 
Demonstration  of   Maxwell's  Law, 
406.2,  cp.  lecture  demon- 
strations. 
Density  determination,  309 

tables,  water,  309;  mercury,  311 

and  v.,  412.5,  425.5  + 
Dextrine,  280,  381.1 
Diastase,  347.3 
Dielectrics,  350.4,  381.5 
Diet,  effect  on  v.  of  blood,  285 
Diffusion,  188,  189,  214,  252,  360.8, 

380.8,  387.2,  390.6,    402.8, 

405.9,  419.9,  421.4 
Diffusional  v.,  147,  150,  242 

Disk  method  of  v.  measurement,  86 
Displacement  of  particles,  400.3 
Dissipation  function,  401.1 
Dissociation,  9,  161,  169,  184,  187, 

195,  413.9 
Double-bond,  118 


Dough,  380.5 
Drainage,  66 
Dynamical  theory,  396.6,  410.3 

E 

Eddies,  14,  39,  42 

Effusion,  241 

Elastic  after-effect,  237,  355.9, 
358.2  388.7,  389.5+, 
390.9,  392.5,  398.5,  399.2, 

401.5,  406.2,  408.5,  414.6, 

425.6,  427.5 
deformation,  4,  212,  217,  218, 

350.2,  353.4,  357.1,  361.7, 

365.5,  369.8,  375.4,  387.3, 

400.9,  401.7,  412.9,  421.5, 

424.9 

limit,  211,  237 
Electric  field,  v.  in  an,  34,  368.6, 

404.4,  408.3,  413.3,  415.6 
Electronic  charge  and  Stoke's  Law, 

411.8,  421.1 
Electroosmosis,  371.9 
Emulsions,  83,  89,  94,  100,  102,  210, 

350.8,  354.8,  356.4,  367.9, 

396.1,405.8,410.8 
End    correction,    21,    et   seq.,    315, 

353.2 

Engine  grease,  211 
Enzyme  reactions,  347.4 
Ethane,  403.6 
Ethers,     113,    364.4,    374.1,    381.8, 

407.1 

Ether-alcohol  mixtures,  350.7 
Ethyl  acetate,  366.3 

alcohol,  361.2,  366.3,  425.1 
water  mixtures,  341 
Ethylene,  403.6,  428.9 
Expansion,  thermal,  65,  363.5 
External     friction,     372.4,     376.2, 

388.4,  389.2 
Eye  fluids,  360.5,  393.7,  396.5 


Falling  sphere,  cp.  sphere. 
Ferric  hydroxide,  407.3 


434 


INDEX 


Filling  viscometer,  310,  312 

Films  as  plastic  substances,  255 

Fineness  of  grain,  235 

First  regime,  39 

Flashing,  39 

Flotation,  361.7 

Flour,  394.4  + 

Flow  through  orifices,  233,  234,  cp. 

hydraulics, 
in  thin  films,  380.2 
of  metals,  235,  236 
theory,     365.2,     403.7,     411.3, 

418.6,  419.3,  423.7,  427.7 
Fluid  defined,  215 
Fluidity  definition,  5,  364.9 

table  for  reference,  318 

in     a     magnetic     field,     cp. 

magnetism, 
in  an  electrostatic  field,  cp. 

electric  field. 
Foam,  211,  229,  409.4 
Formamide,  368.5,  397.9 
Free  volume,  142 
Friction,  cp.  yield  value,  238,  262, 

280,  392.3,  415.9,  428.5 
Fused     salts,     193,     371.7,     374.5, 
386.6,  394.1,  406.4 


Gases,  241,  242,  351.4,  355.9, 
358.4,  360.9,  361. 1+, 
362.6,  367.3  +  ,  368. 1  +  , 
371.1,  374.9,  377.4,  384.9, 
385.3,  388.6,  389.1,  392.2, 
398.3,  403.5,  404.2,  410.8, 

414.3,  419.8,       420.6+, 

422.1,  424.5  +  ,       426.6, 
428.7 

Gasoline,  380.9 

Gelatine,  198,  212,  280,  289,  349.1, 

352.4,  355.7,  375.8,  380.8, 
384.6,  392.9,  393.6,  394.3, 

400.2,  412.8,  414.5,  415.9, 
419.1 

Geophysics,  287 

Glass,  286,  377.9,  384.7,  392.6, 
418.1,  424.6,  425.9 


Glue,  280,  355.7,  370.4 

Gluten,  280,  419.1 

Glycerol,  358.9,  413.3,  414.1  + 

Gold  value,  393.2 

Graphite,  229 

Greases,  281,  361.8,  395.3 

Gums,  419.' 


Haemodynamics,  401.9 

Halogens,  250 

Hardness,  3,  235,  391.4 

Heat  of  vaporization  and  v.,  372.9 

of  fusion,  378.2 
Hemoglobin,  356.8 
Helium,  364.5,  403.5,  409.3,  413.7, 

414.8,  420.2 
Heptane  as  a  standard  for  vapor 

pressure  comparisons,  157, 


Hexamethylene,  277 

High  temperature  v.,  349.1,  370.1 

Homogenizing,  211,  281. 

Hydrate  theory,  354.8 

Hydraulic  flow  and  plastic  state,  231 

Hydraulics,  cp.  also  turbulence, 
365.1,  369.9,  371.6,  372.3, 
380.6,  394.2,  395.3+, 

398.5,  400.1,       407.7  +  , 
408.9,  412.2,  417.4,  421.4 

Hydrocarbons,  113,  351.3,  372.8 
Hydrocellulose,  353.2 
Hydrodynamics,      1,     212,     351.9, 

353.6,  356.6,  358.1,  362.9, 

368.8,  369.1,         373.9, 
391.9+,       394.1,       395.9, 
399.5,  401.3,  401.8,  406.7, 
410.3,       415.9,       418.3+, 
424.3,  426.6 

Hydrogen,      353.5,      376.8,   389.8, 

395.9,  403.5,  408.1,  409.3, 
424.5,  428.4 

bromide,  397.3 
chloride,  397.3 
iodide,  397.3 
sulphide,  397.3 


INDEX 


435 


Hydrogenation,  281 
Hydrolysis,  359.4,  362.2 
Hysteresis,     elastic,     360.3,     368.9, 
391.5 


Ice,  239,  363.7,  371.6,  381.5,  395.4, 

397.2,  400.7,  405.6,  422.9, 

424.2 

Ideal  mixtures,  162 
Immiscible  liquids,  87,  211 
Inflection  curves,  178 
Infusorial  earth,  230 
Interrupted  flow,  28,  60 
Iodides,  114 
Ionic  size,  357.1,  391.6 
Ionic  mobility,  358.3,  395.1,  381.7, 

427.2 

lonization,  195,  350.9,  394.9 
Iron  and  steel,  348.2,  375.4,  388.7, 

407.6,  410.5 

Iso-grouping,  108,  117,  125,  144 
Isothermals  of  fluidity,  146 


K 


Kaolinite,  385.4 

Kinetic    energy,    2,   17,  et  seq.,  59, 
373.5,384.9,  385.2,  420.1 


Law  of  Batschinski,  142,  247 

Poiseuille,    8,    et    seq.,    365.4, 

367.9,  374.2,  375.8,  377.2, 

386.8,  400.9,  403.1,  406.5, 

409.7,  427.1 

Stokes,  188,  402.2,  411.8 
Lard  oil  as  a  cutting  oil,  270 
Lava,  287 
Lecture       demonstrations,      397.8, 

410.9 

Lime,  281,  388.4 
Limiting  volume,  142 
Linear  flow,  410.2 
Liquid  mixtures,    363.4+,  364.4  +  , 


366.1,  367.2+,       368.6, 
369.7,  370.8,  373.6,  387.1, 

387.5,  391.3,  392.7,  393.5, 
397.9,  398.2,  402.2,  412.3, 

413.6,  415.7,  421.3,  422.7 
Liquids,  359.4,  361.5,  374.8,  379.9, 

386.2,  398.3,       400.4+, 
405.2,  406.9,  421.1,  426.3 

Lithium  chloride,  195,  374.6,  383.4 

nitrate,  386.1 
Logarithmic  decrement,  236 

homologues,  45 

viscosities,  104 
Lubricant,  air  as  a,  388.2 
Lubricants,     367.9,     384.1,     400.1, 

412.8 

Lubricating  oils,  370.9,  386.8, 
387.9,  388.5,  390.5,  391.8, 

394.4,  395.6,  401.6,  401.9, 
42.9,  406.8 

value,  269 

Lubrication,  261,  264,  et  seq.,  347.8, 
347.9,  348.2,  4,  5,  6,  9, 
353.2,  363.6,  369.2,  377.1, 
378.9,  379.1,  394.8,  396.2, 
399.2,  405.3,  409.2,  410.2, 
416.6,  417.4,  417.6,  419.6, 

421.5,  421.9,  423.1,  426.7, 
426.8 

and  adhesion,  268 


M 


Magnetism,  34,  350.1,  351.4,  360.4, 
375.4,  389.3,  394.7,  401.6, 
421.8 

Manometer,  307 

Marble,  flow  of,  347.5 

Marine  glue,  235 

Mass  of  hydrogen  atom,  253 

Mayonnaise,  211 

Mean  free  path,  243 

Measurement  of  high  v.,  378.3 

Medicine,  284 

Melting  point  of  tars,  415.8 

Menthol,  235 

Mercury,  352.1,  358.4,  361.9,  368.3, 


436 


INDEX 


388.4,  389.2,  415.4,  416.7, 

423.5,  424.6    425.5,   426.9 
stabilizer,  294 

vapor,  388.4,  402.4 
Metal  ammonia  salts,  355.1 
Metals,  348.1,    377.5,  383.2,  384.4, 

385.4,  387.4,  394.3,  398.6, 
404.9,  410.8,  416.9,  417.9, 

419.6,  421.7,  424.9,  426.8 
Metallurgy,  284 

Methyl  chloride,  130,   171,  371.3 
Methylene  group,  117,  123 
Migration  velocity,  185,  191 
Milk,  284,  286,  351.8,  359.6,  359.7, 

360.5,  372.4,  377.4,  389.2, 
389.8,  390.2,  394.6,  395.2, 
403.1,404.2,423.3 

Mixtures,  84,  90,  et  seq.,  251,  349.7, 
354.1,  cp.  liquid  mixtures. 
Mobility,  217,  218,  219,  220,  221, 

226,  257,  280 
of  ions,  356.3 
Mobil  oil  BB,  141 
Molecular  attraction  and  viscosity  of 

gases,  246 

limiting  volume,  142,  144 
viscosity  work,   107,   109,   110, 

111 
volume,  392.5 

inner  and  outer,  145 
Mortars,  368.2 
Motor  fuels,  400.1 
Multiviscometer,  341.5 


N 


397.6,  402.1,  405.9,  415.1, 
cp.  cellulose  esters. 

Nitrogen,  353.5,  354.7,  395.9,  400.6, 
428.4 

Nomenclature,  7 

Non-electrolytes,  400.7 

Normal  mixtures,  81 


Ohm's  Law,  83 
Oil,  films  on  water,  255 
Oiliness,  269 

Oils,    360.1,    360.4,    361.2,    362.1, 
364.9,  366.6,  366.9,  368.4, 
370.2,  372.9,  373.8,  380.5, 
381.9,       382.7  +  ,       383.9, 
399.6,  401.8,  405.6,  408.5, 
412.4,  412.6,  417.7,  418.8, 
419.6,  423.5,  424.8 
blown,  395.6 
essential,  385.2 
fish,  427.2,  etc. 
fixed,  406.3 
flow  in  pipes,  407.3 
from  Oklahoma  t>s.  Pa.,  404.7 
mixtures,  415.2 
on  metals,  409.6 
Olive  oil,  130,  359.3,  410.2 
Opalescence,  94,  411.8 
Orifices,  363.1 

Ortho  phosphoric  acid,  417.1 
Oscillatory  motion,  397.7 
Oxycellulose  nitrate,  353.2 
Oxygen,     124,     144,    353.5,    395.9, 
428.4 


Naphthenic  acids,  408.2 

Negative  v.  and  negative  curvature, 

160,    169,    178,   etc.,    183, 

etc.,    3734,    386.1,    420.4, 

etc. 

Nickel,  372.7,  375.4 
Nitric  acid,  361.2 
Nitrobenzene,  420.7 
Nitrocellulose,     280,     291,     350.7, 

373.7,  382.1,  393.2,  396.3, 


Paint,  222,  282,  354.4,  400.2,  411.1 
Pastes,  360.3,  381.1,  396.6 
Pendulum    method,     2,     6,     350.6, 

374.2,376.1,398.5 
Penetrance,  259,  351.9,  414.1 
Pepsin  action,  417.8 
Periodic  relationships,  185,  250 
Pharmacy,  284 
Phenol,  412.7,  420.6 


INDEX 


437 


Phosphine,  397.3 

Pigments,  282 

Pitch,  216,  235,  406.4,  422.6,  423.9, 

424.2 

Plasma,  347.4,  426.4 
Plastic  flow,  4,  52,  228,  254 

measurement,  321 
Plasticity,   215,  etc.;   349.3,   349.8, 

350.1,  354.3,    355.3,    etc.; 

356.2,  390.3,  358.5,  358.6, 
359.5,  362.7,  363.1,  367.7, 
368.2,  376.5,  388.2,  388.6, 
390.5,  391.3,  392.8,  399.3, 
401.2,  403.7,  405.2,  406.4, 
409.7,  411.2,  417.3,  418.9, 
423.7,  427.1,  429.1 

and  bacteria,  419.5 

calculation,  323 

of  clay,  387.8 

definition,  216 

and  fusibility,  411.9 

of  ice,  239,  cp.  ice. 

of  salt  rocks,  391.6 

series  of  metals,  236 

and  solubility,  293 

of  steel  and  glass,  392.6 
Plastics,  368.2,  374.2 
Plastometer,  375.5,  406.8 
Plate  glass  flow,  39 
Pleural  exudate,  390.3 
Poise,  61 

Polar  colloids,  208,  212 
Polarization  and  fluidity,  35 
Polydispersed  systems,  394.4 
Positive    curvature    and    chemical 

combination,  172,  183 
Potassium  bromide  solutions,  182 

halide  solutions,  381.2 

iodide,  373.5,  374.6 

nitrate,  374.6 

thiocynate,  374.6 
Precipitation     of     colloids,     380.7, 

384.5,  384.6 

Pressure,  138,  et  seq.,  243,  351.6, 
354.2,  361.8,  369.7,  372.5, 
379.7,  384.1,  385.5,  391.7, 
410.9,  418.3 


Pressure,  corrections,  299 
regulation,  294,  305 
true  average,  298 

Proteins,  356.7,  361.3,  373.1,  393.6, 
399.2,  400.3,  404.9,  410.7, 
423.6,  426.4 
Pseudoglobulin,  361.3 
Pyridine,  379.3 


Quartz,  viscosity  of,  377.6 
R 

Racemic  compounds,  112,  366.8 

Radius  of  capillary,  315 

Raffinose  solutions,  426.2 

Rare  gases,  250,  378.8 

Rate    of    crystallization,    190,    cp. 

solidification, 
of  hydration,  410.7 
of  reaction,  366.4,  376.3 

Reciprocal  properties,  83 

Refractive  index,  393.8 

Regimes,  4,  142 

Relaxation  number,  128 

Residual  affinity,  112 

Resistance,  cp.  conductivity. 

Reynolds  critical  velocity,  40 

Rigidity,  128,  218,  256,  384.4,  398.1 

Ring  grouping,  124 

Road  building,  282 

Rocks,  352.8,  353.3 

Roentgen  rays  as  affecting  viscosity, 
410.1 

Roughness  of  surfaces,  149 

Rubber,  212,  280,  350.3,  352.6, 
353.3,  371.5,  372.9,  388.3, 
394.5,  410.4,  406.7,  411.5, 
etc.,  413.6,  418.7,  423.4, 
428.2 

Rupture,  229 

S 

Sagging   beam    method,    plasticity, 

227 
Salt  solutions,  347.2,  348.7,  349.4+, 


438 


INDEX 


355.1,  359.1,  363.2,  368.7, 

374.6,  376.4,  376.6,  378.4, 

379.7,  381.2,  383.4,  383.7, 

384.8,  385.6,  386.7,  396.3, 

399.9,  402.3,  408.6,  412.2, 
418.1,     420.5,     425.2     +, 

427.5,  403.6 

Saponine,  254,  401.5,  418.1 
Saybolt  Universal  Viscometer,  324, 

etc. 

Scums,  256 
Sealing   wax    as    a    viscous    liquid, 

216,  235 
Seawater,  390.4 

Second  regime,  see  turbulent  flow. 
Seeding,  272 
Seepage,  213,  223,  231 
Separation  of  components  of  mixture 

by  flow,  257,  258,  259 
Serum,  393.4,  396.9,  397.7,  411.6 
Settling  of  suspensions,  188 
Shales,  viscosity  of,  359.1 
Shear,  viscosity  at  low,  365.3 
Sheet  glass  flow,  39 
Shifting    of    minimum    in    fluidity, 

cone,  curves,  174 
Silicate    melts,    287,    364.3,    370.3, 

375.7,  393.8 

Silver  nitrate,  181,  374.6 
Size  of  molecules,  367.8,  384.9 

of   particles   in   colloid,    365.9, 

380.6,  419.9,  428.1 
Slags,  287,  370.3 

Slipping,   14,  29,  et  seq.,   148,  223, 
225,  231,  244,  378.4,  380.4, 

395.7,  425.5,  427.2 
and  superficial  fluidity,  256 

Slip,  367.3 

Soap    solutions,    254,    291,    357.1, 

369.6,374.4,396.9,397.1 
Sodium  chloride,  394.7 

hydroxide,  357.3 

nitrate,  374.6 

salt  solutions,  of  organic  acids, 

392.4 

Softening  temperature,  133 
Soil  moisture,  359.5 


Solid,  definition  of,  215 
friction,  262,  373.2 

Solidification  velocity,  190,  420.1, 
427.8 

Solids,  238,  239,  351.5,  353.9, 
358.3,  363.8,  375.3,  375.4, 
377.5,  378.6,  381.8,  407.2, 
409.7,  414.2,  415.6,  416.8, 
420.9,  422.3,  422.5,  424.1, 
425.5 

Solubility    of    glass,    effect    of    on 

viscosity,  377.3 
and  plasticity,  293 

Solutions,  160,  280,  363.1,  400.5, 
410.1 

Sound  and  viscosity,  380.4 

Specific  volume  differences,  164,  165 
heat      and      viscosity,     368.7, 

371.1 

volumes  of  binary  liquid  mix- 
tures, 382.8,  388.8 

Sphere,  falling,  2,  6,  357.9,  362.6, 
373.7,  397.4,  414.4 

Stabilizer,  294 

Standard  substances,  354.6 

Stannic  chloride,  391.2 

Starches,  373.4,  395.1,  406.8,  419.1 

Steel,  351.8,  361.3,  392.6 

Stereoisomerism,  420.6 

Stokes'  method,  253,  329,  349.2 

Strain,  235 

Stress,  influence  of  on  properties, 
364.5 

Structure,  198 

Sugar  solutions,  407. 1 

Sulphur,  359.2,  369.6,  395.6,  411.8, 

417.1,  422.2 
dioxide,  371.3 

Sulphuric  acid,  361.2,  366.2,  388.9 

Superficial     fluidity,     254,     357.8, 

401.5,  402.5,   409.4,   414.7 
Surface  films  as  plastic  solids,  255 

tension,  35,   56,   96,    101,  211, 
271,    356.8,    356.9,    359.3, 

371.6,  376.7 

Surtension  and  viscosity,  395.8 
Suspensions,    102,    104,    203,    205, 


INDEX 


439 


350.8,  367.6,  383.9,  385.2, 

399.3,  400.8 

of  sulphur,  402.7 

Sutherland's  equation,  247 

Swelling  of  colloids,  404.2 

Syrups,  371.7,  407.1,  407.6 


Tables,  fluidities  and  viscosities  of 

water,  339,  340 
of  ethyl   alcohol  water  mix- 
tures, 341 

of     sucrose     solutions,     341 
logarithms,  345 
radii  limits  for  capillaries,  318 
radius  corresponding  to  weight 

of  mercury,  316 
reciprocals,  342 
values  of  K,  300 
of  M,  301 
of  N,  303 

"Tackiness,"  411.5 
Tallow  as  a  plastic  solid,  216 
Tars,  415.8 
Tautomerism,  111 
Technical  viscometry,  324 
Temperature,    13,   92,    127,   et  seq., 
238,  245,  304,  350.1,  365.3, 
376.9,  379.9,  409.8 
Tensile  strength  and  plasticity,  235 
Tetraethylammoniumiodide,  194 
Textiles,  282 

Third  or  mixed  regime,  35,  42 
Thymol,  413.1 
Time  of  relaxation,  128 
measurement,  304 
Tortion  method,  226,  364.6 
Traction  method,  226 
Tragacanth,  394.5 
Transition  points,  112,  293,  366.4 
Transpiration,  2,  6,  241 
Trypsin,  352.5,  424.4 
Turbulence,   4,   35,   51,   97,   356.5, 
357.9,    364.7,    371,    386.9, 
388.1,  392.6,  399.2,  411.9, 
412.9,  415.7,  417.5,  427.8 
Turpentine,  53,  273 


U 

Ultimate  electric  charge,  252 
Undercooled  liquids,  420.1 
Unsaturation,  366.8 
Urea,  181 
Urethane,  410.4 
Urine,  356.2,  359.9 


Vapor     pressure,     155,     156,     276, 

353.9,  406.9 
Vapors,  246,  398.2,  407.2 +  ,  409.1, 

414.9,  418.7 
Varnish,  358.8,  372.6,  400.2,  415.5, 

423.6,  428.1 
Velocity     of     crystallization,      cp. 

solidification. 
Viscometer,  7 

air  bubble,  350.7 

Barbey,  350.6,  405.1,  412.5 

Clark,  368.2 

constant  pressure,   62,   et  seq., 

404.8,  416.2 

Engler,     350.6,     367.5,     375.5, 

380.9,  389.4,  397.5,  403.3, 
405.1,408.4,423.1 

Fischer,  370.9 

Flowers,  371.4,  381.2 

Gurnbel,  413.8 

Gurney,  377.2 

Lunge,  394.5,  413.6 

MacMichael,  328,  416.1 

Maxwell,  414.2 

Ostwald,  403.8 

Redwood,  397.5 

Saybolt,  324,  380.9,  397.5,  375.5 

Schulz,  414.7 

Searle,  415.5 

Stormer,  410.6,  411.1,  419.5 

Washburn,  426.2 
Viscose,  280 
Viscosity  definition,  5,  378.6 

measurement,  6 

Viscous  liquids,  374.7,  391.7,  399.7, 
402.6,   415.8,  418.2,   423.9 
Volume,  141,  142,  184,  373.5  + 


440  INDEX 

W  Y 

Yield  value,  217,  et  seq.,  237,  257 

Water,   347.6,   351.1,  364.8,  373.3, 

375.8,  383.4,  383.5,  388.9,  Z 

391.4,  395.5,  399.8,  404.3,      Zero  fluidity  concentration,  54,  201, 
408.3,  416.8,  427.4  203,  205,  220 

Whipped  cream,  211  Zinc-cadmium  alloys,  424.9 

Wide  tubes,  397.8  Zinc  sulphate,  394.3 


QC  171 
B 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 

Los  Angeles 
This  book  is  DUE  on  the  last  date  stamped  below. 


Ubr 


f  pp 


i 

AA    000508110    4 


UCLA-Physics  Library 
QC  171  B51f 


L  006  579  223  6 


